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Ultrasound Contrast Agent Modeling: A Review

  • Michel Versluis
    Correspondence
    Address correspondence to: Michel Versluis, Physics of Fluids Group, MESA+ Institute for Nanotechnology, Technical Medical (TechMed) Center, University of Twente, P.O. Box 217, Enschede 7500 AE, the Netherlands.
    Affiliations
    Physics of Fluids Group, MESA+ Institute for Nanotechnology, Technical Medical (TechMed) Center, University of Twente, Enschede, the Netherlands
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  • Eleanor Stride
    Affiliations
    Department of Engineering Science, Institute of Biomedical Engineering, University of Oxford, Oxford, UK
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  • Guillaume Lajoinie
    Affiliations
    Physics of Fluids Group, MESA+ Institute for Nanotechnology, Technical Medical (TechMed) Center, University of Twente, Enschede, the Netherlands
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  • Benjamin Dollet
    Affiliations
    Centre National de la Recherche Scientifique (CNRS), Laboratoire Interdisciplinaire de Physique (LIPhy), Université Grenoble Alpes, Grenoble, France
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  • Tim Segers
    Affiliations
    Physics of Fluids Group, MESA+ Institute for Nanotechnology, Technical Medical (TechMed) Center, University of Twente, Enschede, the Netherlands
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Open AccessPublished:June 13, 2020DOI:https://doi.org/10.1016/j.ultrasmedbio.2020.04.014

      Abstract

      Ultrasound is extensively used in medical imaging, being safe and inexpensive and operating in real time. Its scope of applications has been widely broadened by the use of ultrasound contrast agents (UCAs) in the form of microscopic bubbles coated by a biocompatible shell. Their increased use has motivated a large amount of research to understand and characterize their physical properties as well as their interaction with the ultrasound field and their surrounding environment. Here we review the theoretical models that have been proposed to study and predict the behavior of UCAs. We begin with a brief introduction on the development of UCAs. We then present the basics of free-gas-bubble dynamics upon which UCA modeling is based. We review extensively the linear and non-linear models for shell elasticity and viscosity and present models for non-spherical and asymmetric bubble oscillations, especially in the presence of surrounding walls or tissue. Then, higher-order effects such as microstreaming, shedding and acoustic radiation forces are considered. We conclude this review with promising directions for the modeling and development of novel agents.

      Key Words

      Introduction

      While in the 17th century Robert Hooke had already predicted that we would be able to image the human body with sound (
      • Tyndall J.
      Sound: 8 lectures.
      ), it was not until the 1940s that the first ultrasound scan of soft tissue was realized (
      • Howry D.H.
      The ultrasonic visualization of soft tissue structures and disease processes.
      ). Today, ultrasound is one of the most widely used medical imaging techniques because of its safety and relatively low expense compared with magnetic resonance imaging and X-ray computed tomography. As with the other modalities, ultrasound allows for functional imaging, including Doppler (
      • Evans D.
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      ;
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      • Armstrong J.
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      A high-frequency continuous-wave Doppler ultrasound system for the detection of blood flow in the microcirculation.
      ), vector flow imaging (
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      ) and neuroimaging (
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      ). Moreover, ultrasound machines are small and portable and can be used at the bedside. Finally, ultrasound provides real-time imaging, offering typically 50 frames per second for conventional line-by-line scanning and up to 10,000 frames/s for ultrafast plane wave imaging (
      • Tanter M.
      • Fink M.
      Ultrafast imaging in biomedical ultrasound.
      ).
      The best-known application of medical ultrasound is fetal scanning. The amniotic fluid around the fetus contains very few scatterers in comparison to the fetal tissue, and as a result the image of the fetus stands out clearly against a black background. Such high contrast can also be found in echocardiography, where the blood in the left ventricular cavity is a poor ultrasound scatterer as compared with the cardiac muscle tissue. This high contrast then aids in the delineation of the endocardial border and the diagnosis of regional wall motion abnormalities (
      • Mulvagh S.L.
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      American Society of Echocardiography consensus statement on the clinical applications of ultrasonic contrast agents in echocardiography.
      ;
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      • Becher H.
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      • Agati L.
      • Zamorano J.
      • Vanoverschelde J.L.
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      Contrast echocardiography: Evidence-based recommendations by European Association of Echocardiography.
      ). However, the poor backscatter from the blood pool is problematic for perfusion imaging and blood flow quantification.
      Ultrasound contrast agents (UCAs) can dramatically boost the native contrast of the blood pool. A UCA consists of a suspension of stabilized gas microbubbles (
      ) that is injected intravenously. As the microbubbles are smaller than red blood cells, they can traverse even the smallest capillaries. Despite their diminutive size, their scattering cross section can be up to nine orders of magnitude larger than that of red blood cells, and this highly efficient scattering enables the quantification of the perfusion of the myocardium and other organs, including liver, brain and kidney.
      The potential of microbubbles as UCAs was in fact discovered by accident in 1968, during an intravenous injection of a saline solution while performing a cardiac ultrasound examination (
      • Gramiak R.
      • Shah P.M.
      Echocardiography of the aortic root.
      ). The microbubbles that were formed in solution by hand agitation were found to scatter the administered ultrasound highly efficiently. Given the poor stability of the air-filled bubbles in the electrolyte solution, considerable effort was put into finding more stable alternatives. This led to the first class of agents—including Quantison (
      • Frinking P.J.A.
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      Acoustic modeling of shell-encapsulated gas bubbles.
      ), Albunex (
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      Two-dimensional contrast echocardiography. I: In vitro development and quantitative analysis of echo contrast agents.
      ) and Optison (
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      • Price R.J.
      • Skalak T.C.
      • Kaul S.
      Hemodynamic characteristics, myocardial kinetics and microvascular rheology of FS-069, a second-generation echocardiographic contrast agent capable of producing myocardial opacification from a venous injection.
      ;
      • Cohen J.L.
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      • Segar D.S.
      • Gillam L.D.
      • Gottdiener J.S.
      • Hausnerova E.
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      Improved left ventricular endocardial border delineation and opacification with OPTISON (FS069), a new echocardiographic contrast agent: Results of a phase III multicenter trial.
      )—containing coated microbubbles that were formed by sonication of human serum albumin. With their relatively rigid shells and corresponding decreased scattering response, a second generation of UCAs was developed that was inspired by the more flexible nature of lipid cell membranes. These novel agents—including SonoVue (
      • Schneider M.
      Characteristics of SonoVue.
      ), Definity (
      • Unger E.
      • Shen D.
      • Fritz T.
      • Kulik B.
      • Lund P.
      • Wu G.L.
      • Yellowhair D.
      • Ramaswami R.
      • Matsunaga T.
      Gas-filled lipid bilayers as ultrasound contrast agents.
      ) and Sonazoid (
      • Sontum P.C.
      Physicochemical characteristics of Sonazoid, a new contrast agent for ultrasound imaging.
      )—are composed of a suspension of purified phospholipid-mixture–encapsulated gas microbubbles, with diameters ranging from 1 to 10 μm, that are filled with a high-molecular-weight gas, typically perfluorocarbons or sulfur hexafluoride. Examples of UCA microbubbles are given in Figure 1a–d.
      Fig 1
      Fig. 1Ultrasound contrast-agent microbubbles. (a) Lipid-coated microbubbles as seen in bright-field microscopy (reprinted with permission from
      • Segers T.
      • de Rond L.
      • de Jong N.
      • Borden M.
      • Versluis M.
      Stability of monodisperse phospholipid-coated microbubbles formed by flow-focusing at high production rates.
      . Copyright 2016 American Chemical Society); (b) perfluoropropane bubbles with a protein shell (Optison), seen here against a background of red blood cells (reproduced from
      • Wilson S.R.
      • Burns P.N.
      Microbubble-enhanced US in body imaging: What role?.
      ); (c) lipid-coated microbubbles of perfluoropropane gas (Definity) as seen in dark-field microscopy (reproduced from
      • Wilson S.R.
      • Burns P.N.
      Microbubble-enhanced US in body imaging: What role?.
      ); (d) scanning electron micrograph of a Quantison microsphere, fractured owing to vacuum (reproduced with permission from
      • Frinking P.J.A.
      • de Jong N.
      Acoustic modeling of shell-encapsulated gas bubbles.
      ); (e) vapor bubble release from a laser-heated single oil-filled microcapsule captured under ultrahigh-speed imaging (reproduced with permission from
      • Lajoinie G.
      • Gelderblom E.
      • Chlon C.
      • Böhmer M.
      • Steenbergen W.
      • de Jong N.
      • Manohar S.
      • Versluis M.
      Ultrafast vapourization dynamics of laser-activated polymeric microcapsules.
      ); (f) scanning electron micrograph of monodisperse microcapsules (image courtesy of Nanomi, Oldenzaal, the Netherlands); (g) monodisperse microbubbles with a radius of 2.7 μm produced in a flow-focusing device (reprinted with permission from
      • Segers T.
      • de Rond L.
      • de Jong N.
      • Borden M.
      • Versluis M.
      Stability of monodisperse phospholipid-coated microbubbles formed by flow-focusing at high production rates.
      . Copyright 2016 American Chemical Society); (h) numerical simulation of the buckling instability of the elastic shell of a polymeric hard-shelled microcapsule (reproduced with permission from
      • Marmottant P.
      • Bouakaz A.
      • de Jong N.
      • Quilliet C.
      Buckling resistance of solid shell bubbles under ultrasound.
      ); (i) buckling instability upon dissolution for two different surfactants (Survanta and dipalmitoylphosphatidylcholine (DPPC)) (reprinted with permission from
      • Thomas A.
      • Borden M.
      Hydrostatic pressurization of lung surfactant microbubbles: Observation of a strain-rate dependent elasticity.
      . Copyright 2017 American Chemical Society); (j) buckling observed during compression of a lipid-coated microbubble (reproduced with permission from
      • Sijl J.
      • Overvelde M.
      • Dollet B.
      • Garbin V.
      • de Jong N.
      • Lohse D.
      • Versluis M.
      “Compression-only” behavior: A second-order nonlinear response of ultrasound contrast agent microbubbles.
      ).
      The coating of the microbubbles decreases (or potentially eliminates) the interfacial tension, which stabilizes the bubble against Laplace pressure–driven dissolution (
      • Epstein P.S.
      • Plesset M.S.
      On the stability of gas bubbles in liquid-gas solutions.
      ). The stability of the microbubbles is further enhanced by both the low solubility of the gas and the resistive barrier that the encapsulation forms against gas diffusion through the monolayer membrane (
      • Borden M.A.
      • Longo M.L.
      Dissolution behavior of lipid monolayer-coated, air-filled microbubbles: Effect of lipid hydrophobic chain length.
      ). Long-chain polyethylene glycol spacers are added to provide stealth capability against detection by the body's immune system and therefore improve the circulation lifetime in vivo (
      • Klibanov A.L.
      Microbubble contrast agents: Targeted ultrasound imaging and ultrasound-assisted drug-delivery applications.
      ). Furthermore, they prevent bubble coalescence through repulsive intermolecular forces (
      • Segers T.
      • Lohse D.
      • Versluis M.
      • Frinking P.
      Universal equations for the coalescence probability and long-term size stability of phospholipid-coated monodisperse microbubbles formed by flow-focusing.
      ;
      • Borden M.A.
      • Song K.H.
      Reverse engineering the ultrasound contrast agent.
      ). Cumulatively, this leads to an overall bubble lifetime of 5–10 mins after intravenous injection (
      • Kaul S.
      Current status of contrast echocardiography.
      ;
      • Lindner J.R.
      Principles of contrast echocardiography.
      ). Perfluorocarbon gas-filled microbubbles can be stable in suspension for several days to a few weeks when kept in a vial under a protective gas headspace. Freeze-dried bubbles can be stored for several years (

      European Medicines Agency. SonoVue: EPAR—Product information. Available at: https://www.ema.europa.eu/en/documents/product-information/sonovue-epar-product-information_en.pdf. Accessed January 22, 2020.

      ).
      The physical mechanism by which microbubbles enhance image contrast is twofold. First, acoustic waves are scattered by the bubbles owing to the large difference in acoustic impedance between the gas core and the surrounding liquid. Second, and more important, the large compressibility of the gas core leads to volumetric oscillations in response to ultrasound pressure waves. Thus, the oscillating bubble acts as a source of sound with an effective scattering cross section up to several orders of magnitude larger than that of a rigid sphere of the same size and acoustic impedance (
      • Hilgenfeldt S.
      • Lohse D.
      • Zomack M.
      Response of bubbles to diagnostic ultrasound: A unifying theoretical approach.
      ). Furthermore, when the bubbles are driven at resonance, their oscillations can be highly non-linear and therefore they can produce scattered ultrasound fields containing a range of frequencies, including whole and fractional harmonics of the driving frequency. In addition, and particularly at lower driving pressures (<100 kPa), further non-linearities are induced by the non-linear viscoelastic properties of the stabilizing shell. To date, contrast-specific imaging modes use the non-linear echoes to detect the contrast agent response while suppressing the linear tissue echoes (
      • Hope Simpson D.
      • Chin C.T.
      • Burns P.N.
      Pulse inversion Doppler: A new method for detecting nonlinear echoes from microbubble contrast agents.
      ;
      • Frinking P.J.A.
      • Bouakaz A.
      • Kirkhorn J.
      • Ten Cate F.J.
      • de Jong N.
      Ultrasound contrast imaging: Current and new potential methods.
      ;
      • Rafter P.
      • Phillips P.
      • Vannan M.A.
      Imaging technologies and techniques.
      ;
      • Eckersley R.J.
      • Chin C.T.
      • Burns P.N.
      Optimising phase and amplitude modulation schemes for imaging microbubble contrast agents at low acoustic power.
      ;
      • Averkiou M.A.
      • Mannaris C.
      • Bruce M.
      • Powers J.
      Nonlinear pulsing schemes for the detection of ultrasound contrast agents.
      ). By fortunate coincidence, the range of microbubble sizes appropriate for clinical use corresponds to resonance frequencies on the order of 1–10 MHz, the typical frequency range used in diagnostic ultrasound. It is this combination of scattering efficiency and non-linearity that makes microbubbles such excellent UCAs.
      While the use of coated bubbles as a perfusion contrast agent is clinically established, the interest in oscillating bubbles as a theranostic agent is also rapidly increasing for the following reasons. First, transfer of momentum to the surrounding fluid by the non-linearly oscillating bubble can set up a steady flow, or microstreaming, that can both affect the transport of therapeutic material and impose shear stresses on biological membranes, thereby altering their permeability and thus mediating therapeutic effects (
      • Ward M.
      • Wu J.
      • Chiu J.F.
      Experimental study of the effects of Optison concentration on sonoporation in vitro.
      ;
      • Marmottant P.
      • Hilgenfeldt S.
      Controlled vesicle deformation and lysis by single oscillating bubbles.
      ;
      • Pereno V.
      • Aron M.
      • Vince O.
      • Mannaris C.
      • Seth A.
      • de Saint Victor M.
      • Lajoinie G.
      • Versluis M.
      • Coussios C.
      • Carugo D.
      • Stride E.
      Layered acoustofluidic resonators for the simultaneous optical and acoustic characterisation of cavitation dynamics, microstreaming, and biological effects.
      ). Second, absorption of the acoustic emissions from the bubbles can produce localized heating of the surroundings (
      • Prosperetti A.
      Thermal effects and damping mechanisms in the forced radial oscillations of gas bubbles in liquids.
      ;
      • Coussios C.C.
      • Roy R.A.
      Applications of acoustics and cavitation to noninvasive therapy and drug delivery.
      ) that can directly produce bio-effects or be used, e.g. to release drugs from thermally sensitive carriers. Heating is increased compared with direct ultrasound exposure on account of the higher-frequency components in the bubble emissions that are more readily absorbed by tissue (
      • Hilgenfeldt S.
      • Lohse D.
      • Zomack M.
      Sound scattering and localized heat deposition of pulse-driven microbubbles.
      ). Third, the increased temperature and pressure in the core of the bubble during compression can stimulate sonochemistry (
      • Suslick K.S.
      • Flannigan D.J.
      Inside a collapsing bubble: Sonoluminescence and the conditions during cavitation.
      ), leading to the production of chemical species including oxygen radicals, although this effect is reduced for the polyatomic gases typically used in contrast agents. Finally, there may be complex interactions between oscillating bubbles and biological structures that promote therapeutic effects. These include microjetting and radiation force phenomena.
      In the mid-1990s, a new field emerged that focused on the complex interaction between coated microbubbles and ultrasound. Advances were made through experimental observations, first using acoustic spectroscopy (
      • de Jong N.
      • Hoff L.
      • Skotland T.
      • Bom N.
      Absorption and scatter of encapsulated gas filled microspheres: Theoretical considerations and some measurements.
      ) and later using ultra high-speed imaging (
      • Morgan K.
      • Allen J.
      • Dayton P.
      • Chomas J.
      • Klibanov A.
      • Ferrara K.
      Experimental and theoretical evaluation of microbubble behavior: Effect of transmitted phase and bubble size.
      ;
      • Chin C.T.
      • Lancée C.
      • Borsboom J.
      • Mastik F.
      • Frijlink M.E.
      • de Jong N.
      • Versluis M.
      • Lohse D.
      Brandaris 128: A digital 25 million frames per second camera with 128 highly sensitive frames.
      ). Along with the experimental work, great progress has been made in model development. The majority of the available coated-bubble models have in common a starting point of a Rayleigh Plesset–type equation (
      • Lauterborn W.
      Numerical investigation of nonlinear oscillations of gas bubbles in liquids.
      ;
      • Matula T.J.
      Inertial cavitation and single-bubble sonoluminescence.
      ). The field of UCA modeling has greatly benefited from the field of sonoluminescence, where the dynamics of free gas bubbles in an unbounded fluid have been fully described (
      • Brenner M.P.
      • Hilgenfeldt S.
      • Lohse D.
      Single-bubble sonoluminescence.
      ;
      • Lohse D.
      Bubble puzzles: From fundamentals to applications.
      ). The effect of the viscoelastic shell on bubble dynamics is evident from acoustical and optical characterization. The pressure contributions resulting from the viscoelastic shell are incorporated into the Rayleigh–Plesset (RP) equation as additional interfacial pressure terms (
      • de Jong N.
      • Cornet R.
      • Lancée C.T.
      Higher harmonics of vibrating gas-filled microspheres, part one: Simulations.
      ;
      • Church C.C.
      The effect of an elastic solid surface layer on the radial pulsation of gas bubbles.
      ;
      • Hoff L.
      • Sontum P.C.
      • Hovem J.M.
      Oscillations of polymeric microbubbles: Effect of the encapsulating shell.
      ). The shell elasticity increases the stiffness of the system, and an increase in the resonance frequency is expected. Similarly, the shell viscosity leads to energy dissipation which results in increased damping and hence a small reduction in resonance frequency and decreased amplitude of oscillation.
      The non-linear properties of the bubble shell are especially important at small oscillation amplitudes. The densely packed lipid monolayer may buckle during compression, while the bubble coating may rupture upon expansion, forming lipid domains on the otherwise free interface and thereby exposing the gas directly to the surrounding liquid. Such changes in the interfacial properties of the coating lead to surprising contrast response characteristics, including thresholding behavior (
      • Emmer M.
      • van Wamel A.
      • Goertz D.E.
      • de Jong N.
      The onset of microbubble vibration.
      ), compression-only behavior (
      • de Jong N.
      • Emmer M.
      • Chin C.T.
      • Bouakaz A.
      • Mastik F.
      • Lohse D.
      • Versluis M.
      “Compression-only” behavior of phospholipid-coated contrast bubbles.
      ;
      • Sijl J.
      • Overvelde M.
      • Dollet B.
      • Garbin V.
      • de Jong N.
      • Lohse D.
      • Versluis M.
      “Compression-only” behavior: A second-order nonlinear response of ultrasound contrast agent microbubbles.
      ) and subharmonics (
      • Sijl J.
      • Dollet B.
      • Overvelde M.
      • Garbin V.
      • Rozendal T.
      • de Jong N.
      • Lohse D.
      • Versluis M.
      Subharmonic behavior of phospholipid-coated ultrasound contrast agent microbubbles.
      ). In addition, gas loss through diffusion, or acoustic deflation, upon ultrasound driving may lead to a higher concentration of lipids at the bubble interface, changing its dynamics. Alternatively, the bubble may expel excess lipids (
      • Luan Y.
      • Lajoinie G.
      • Gelderblom E.
      • Skachkov I.
      • van der Steen A.F.W.
      • Vos H.J.
      • Versluis M.
      • de Jong N.
      Lipid shedding from single oscillating microbubbles.
      ), leading to higher surface tension and promoting gas loss (
      • Viti J.
      • Mori R.
      • Guidi F.
      • Versluis M.
      • de Jong N.
      • Tortoli P.
      Nonlinear oscillations of deflating bubbles.
      ). Altogether, the interfacial rheology is an important factor in fully describing the origin and details of the non-linear echoes generated by bubbles. Ultimate control over the shell properties and its physical behavior is elementary for the design of optimized agents for contrast imaging, displaying improved backscatter, less attenuation and superior non-linear properties. Such agents can then also act as vector flow tracers for echo particle image velocimetry (
      • Leow C.H.
      • Bazigou E.
      • Eckersley R.J.
      • Yu A.C.H.
      • Weinberg P.D.
      • Tang M.-X.
      Flow velocity mapping using contrast enhanced high-frame-rate plane wave ultrasound and image tracking: Methods and initial in vitro and in vivo evaluation.
      ;
      • Poelma C.
      Ultrasound imaging velocimetry: A review.
      ;
      • Engelhard S.
      • Voorneveld J.
      • Vos H.J.
      • Westenberg J.J.M.
      • Gijsen F.J.H.
      • Taimr P.
      • Versluis M.
      • de Jong N.
      • Bosch J.G.
      • Reijnen M.M.P.J.
      • Groot Jebbink E.
      High-frame-rate contrast-enhanced US particle image velocimetry in the abdominal aorta: First human results.
      ) and enable non-invasive pressure measurements (
      • Shi W.T.
      • Forsberg F.
      • Raichlen J.S.
      • Needleman L.
      • Goldberg B.B.
      Pressure dependence of subharmonic signals from contrast microbubbles.
      ;
      • Frinking P.J.A.
      • Gaud E.
      • Brochot J.
      • Arditi M.
      Subharmonic scattering of phospholipid-shell microbubbles at low acoustic pressure amplitudes.
      ).
      In addition to the dramatic effect of the stabilizing shell, bubble dynamics are highly dependent on the biological environment and the presence of neighboring bubbles. A compliant blood vessel wall will reflect part of the pressure field emitted by the bubble in a non-trivial way, resulting in a complex interaction with the bubbles (
      • Garbin V.
      • Cojoc D.
      • Ferrari E.
      • Di Fabrizio E.
      • Overvelde M.L.J.
      • van der Meer S.M.
      • de Jong N.
      • Lohse D.
      • Versluis M.
      Changes in microbubble dynamics near a boundary revealed by combined optical micromanipulation and high-speed imaging.
      ;
      • Doinikov A.A.
      • Zhao S.
      • Dayton P.A.
      Modeling of the acoustic response from contrast agent microbubbles near a rigid wall.
      ;
      • Hay T.A.
      • Ilinskii Y.A.
      • Zabolotskaya E.A.
      • Hamilton M.F.
      Model for bubble pulsation in liquid between parallel viscoelastic layers.
      ;
      • Helfield B.L.
      • Leung B.Y.C.
      • Goertz D.E.
      The effect of boundary proximity on the response of individual ultrasound contrast agent microbubbles.
      ). Similarly, the sound field generated by neighboring bubbles interacts with that of the bubble, thereby altering its dynamics and inducing acoustic radiation forces (
      • Bjerknes V.F.K.
      Fields of force: Supplementary lectures, applications to meteorology.
      ;
      • Crum L.A.
      Bjerknes forces on bubbles in a stationary sound field.
      ). These effects are considered higher-order effects and are discussed in more detail later under Modeling higher-order effects, as well as in the companion review on therapeutic applications of microbubbles (
      • Kooiman K.
      • Roovers S.
      • Langeveld S.A.G.
      • Kleven R.T.
      • Dewitte H.
      • O'Reilly M.A.
      • Escoffre J.-M.
      • Bouakaz A.
      • Verweij M.D.
      • Hynynen K.
      • Lentacker I.
      • Stride E.
      • Holland C.K.
      Ultrasound-responsive cavitation nuclei for therapy and drug delivery.
      ).
      This article reviews the modeling efforts that have been undertaken, starting with bubble fundamentals, bubble shell models and secondary higher-order effects, and finishing with new directions that are explored for improved models to describe bubble dynamics in tissues and those of novel agents such as bubble clusters and bubbles for multimodality imaging. One or more of our cited studies used human subjects, and these were, to the best of our knowledge, granted protocol approval by an ethics committee or institutional review board. Where animals were studied the research protocol was, again to the best of our knowledge, approved by a local institutional care and animal use committee.

      Modeling free-bubble dynamics

      The free-gas-bubble model

      Numerous mathematical formulations have been developed to describe bubble dynamics. Historically these have been motivated primarily by the hydrodynamic effects of bubbles in turbomachinery. The first published use of the term cavitation dates to
      • Thornycroft J.I.
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      Torpedo-boat destroyers.
      and is attributed to the engineer and naval architect Robert Froude. That bubbles might be generated by the motion of a water turbine had in fact been proposed over a century earlier by
      • Euler L.
      Théorie plus complette des machines qui sont mises en mouvement par la réaction de l'eau.
      , but it was not until the late 19th century that the relationship between liquid pressure, bubble formation and, importantly, physical effects on the surroundings became the focus of intense research.
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      and subsequently
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      and
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      considered an empty spherical cavity in a large volume of inviscid fluid. In this case the bubble undergoes a single, very rapid, collapse, driven by the inertia of the surrounding fluid and generating a shock wave as a result of the high pressure developed at the site of the bubble.
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      ,
      • Noltingk B.E.
      • Neppiras E.A.
      Cavitation produced by ultrasonics.
      and
      • Neppiras E.A.
      • Noltingk B.E.
      Cavitation produced by ultrasonics - theoretical conditions for the onset of cavitation.
      subsequently considered the more commonly encountered case of a bubble partially or completely filled with gas. In this case the pressure inside the bubble increases as it is compressed, producing a cushioning effect. Finally, an additional term was included to account for viscous dissipation in the surrounding fluid. According to
      • Young F.
      Cavitation.
      this was originally given by
      • Taylor G.I.
      • Davies R.M.
      The motion and shape of the hollow produced by an explosion in a liquid.
      , although it is generally attributed to
      • Poritsky H.
      The collapse or growth of a spherical bubble or cavity in a viscous fluid.
      . The result was the Rayleigh-Plesset or Rayleigh–Plesset–Noltingk–Neppiras–Poritsky equation (
      • Lauterborn W.
      Numerical investigation of nonlinear oscillations of gas bubbles in liquids.
      ):
      ρL(RR¨+32R˙2)=PBP.
      (1)


      This is a pressure balance between the inertial term on the left-hand side, originating from the inertia of the liquid associated with the bubble motion, and the pressure difference on the right-hand side arising from the difference in work done at the bubble wall and done remotely from the bubble (
      • Leighton T.G.
      The inertial terms in equations of motion for bubbles in tubular vessels or between plates.
      ). Table 1 summarizes all symbols used in this review. In this equation, R is the time-dependent bubble radius (see Fig. 2, left), with initial value R0; R˙ and R¨ are the velocity and acceleration, respectively, of the bubble wall; ρL is the density of the surrounding fluid; PB is the pressure in the liquid at the bubble wall; and P is the pressure in the liquid far away from the bubble, defined as the sum of the hydrostatic pressure P0 and the time-varying driving pressure PA(t) in the liquid.
      Table 1List of symbols
      SymbolMeaning
      ABubble surface area
      A0Bubble surface area at rest
      anAmplitude of surface mode number n
      αTotal attenuation
      cLSpeed of sound in the liquid
      dAcoustic path length
      dABubble surface-area dilatation
      δBLBoundary-layer thickness
      δtotTotal damping
      δradRadiative damping
      δvisLiquid viscous damping
      δshellShell viscous damping
      δthThermal damping
      eBubble location in the tube
      ϵSmall oscillation amplitude (dimensionless)
      ϕAzimuthal angle of the spherical harmonics
      ηBackscatter coefficient
      fUltrasound frequency
      fTTransmit frequency
      fresResonance frequency
      f0Bubble eigenfrequency
      κPolytropic exponent
      κsShell viscosity
      lOne-dimensional bubble length
      l0One-dimensional bubble length at rest
      LOne-dimensional tube length
      Typical streaming length scale
      μLLiquid viscosity
      nSpherical harmonic mode number
      n(R)Bubble concentration of bubbles with radius R
      P0Hydrostatic pressure
      PPressure in the liquid far from the bubble
      PADriving pressure
      PBPressure in the liquid at the bubble wall
      pvVapor pressure
      PsScattered pressure
      PTTransmit pressure
      PvisViscous pressure term
      PelasElastic pressure term
      RBubble radius
      R˙Bubble wall velocity
      R¨Bubble wall acceleration
      R0Bubble radius at rest
      RbBubble buckling radius
      RrBubble rupture radius
      RtubeTube radius
      ρLLiquid density
      σSurface tension
      σ(R)Size-dependent surface tension
      σ(R0)Surface tension at rest
      σsScattering cross section
      tTime
      θPolar angle of the spherical harmonics
      τxzShear stress along the wall
      UOscillatory flow velocity
      usStreaming velocity
      VBubble volume
      ωAngular ultrasound frequency
      ω0Angular bubble eigenfrequency
      ω0nAngular bubble eigenfrequency of surface mode n
      ΩsLinearized scattering cross section
      xCoordinate along the wall
      χShell elasticity
      YnSpherical harmonic of mode number n
      zCoordinate away from the wall
      Fig 2
      Fig. 2Schematic view of a free gas bubble (left) and a lipid-coated microbubble (right). To the right, the coating is indicated by the primary phospholipids and lipids with polyethylene glycol added (not drawn to scale). R0 = bubble size; PB = liquid pressure at the bubble wall; P = liquid pressure far away from the bubble; μL = liquid viscosity; ρL = liquid density; cL = speed of sound in the liquid; σ = surface tension of water; κs = shell viscosity; χ = shell elasticity; σ(R) = size-dependent effective surface tension of the coated bubble.
      When the gas in the bubble is an ideal gas, from the pressure balance at the bubble interface PB can be expressed as
      PB=(P0+2σR0pv(R0))(R0R)3κ2σR4μLR˙R+pv,
      (2)


      following the ideal gas law, with σ the surface tension at the gas–liquid interface, κ the polytropic constant and μL the viscosity of the surrounding liquid. The vapor pressure pv inside the bubble is typically neglected in the modeling of the dynamics of micrometer-sized bubbles because the vapor pressure of water at atmospheric conditions (2.3 kPa) is low as compared with the hydrostatic and interfacial pressure terms.
      There are a number of assumptions implicit in the derivation of eqns (1) and (2): the bubble is assumed to remain spherical at all times, with no heat or mass transfer across the bubble wall during oscillation. The surrounding fluid is assumed to be infinite, incompressible and Newtonian. The gas inside the bubble is assumed to behave ideally. The wavelength of the incident ultrasound field is assumed to be sufficiently large that the bubble experiences a spatially uniform pressure. The effects of gravity and buoyancy are ignored.

      Acoustic reradiation

      At small amplitudes of oscillation the assumption of an incompressible fluid is justified. However, with increasing driving pressure it becomes necessary to modify eqn (1) to correct for the effects of liquid compressibility and energy dissipation associated with generating acoustic emissions. This may become significant for situations in which the velocity of the bubble wall is comparable with the speed of sound in the surrounding liquid (|R˙|/cL1). Based on the work of
      • Herring C.
      Theory of the pulsations of the gas bubble produced by an underwater explosion.
      ,
      • Trilling L.
      The collapse and rebound of a gas bubble.
      rederived eqn (1) to take into account the energy stored in the liquid during bubble oscillation and the damping effect produced by the re-radiation of sound waves. This treatment assumes that the sound velocity cL in the surrounding liquid is constant and is therefore limited in terms of the maximum amplitude of oscillation for which it is valid.

      Gilmore F.R.The growth or collapse of a spherical bubble in a viscous compressible liquid. Hydrodynamics Laboratory Report No. 26-4. Pasadena, CA: California Institute of Technology, 1952.

      improved upon this assumption by employing the Kirkwood–Bethe hypothesis (
      • Bethe H.A.
      Statistical theory of superlattices.
      ;
      • Kirkwood J.G.
      Order and disorder in binary solid solutions.
      ), which states that the wave velocity is equal to the sum of the velocity of the liquid and the local speed of sound at the bubble wall.
      In 1980, building upon the earlier models,
      • Keller J.B.
      • Miksis M.
      Bubble oscillations of large amplitude.
      published a more complete treatment for large-amplitude bubble oscillations, which included surface tension and liquid viscosity, both of which were neglected in earlier work:
      (1R˙cL)RR¨+32(1R˙3cL)R˙2=1ρL(1+R˙cL+RcLddt)(PBP),
      (3)


      with PB and P as defined before. This expression was subsequently refined by
      • Prosperetti A.
      • Crum L.A.
      • Commander K.W.
      Nonlinear bubble dynamics.
      , who removed the assumption of ideal gas behavior and coupled the equation of motion for the bubble to the equations of mass and energy conservation. Note that eqn (3) is no longer valid when |R˙|/cL1. Therefore, a popular form in the context of sonoluminescence and UCA modeling is given by (
      • Löfstedt R.
      • Weninger K.
      • Putterman S.
      • Barber B.P.
      Sonoluminescing bubbles and mass diffusion.
      ;
      • Barber B.P.
      • Hiller R.A.
      • Löfstedt R.
      • Putterman S.J.
      • Weninger K.R.
      Defining the unknowns of sonoluminescence.
      ;
      • Brenner M.P.
      • Hilgenfeldt S.
      • Lohse D.
      Single-bubble sonoluminescence.
      ;
      • Marmottant P.
      • van der Meer S.
      • Emmer M.
      • Versluis M.
      • de Jong N.
      • Hilgenfeldt S.
      • Lohse D.
      A model for large amplitude oscillations of coated bubbles accounting for buckling and rupture.
      ;
      • Paul S.
      • Katiyar A.
      • Sarkar K.
      Material characterization of the encapsulation of an ultrasound contrast microbubble and its subharmonic response: Strain-softening interfacial elasticity model.
      ):
      ρL(RR¨+32R˙2)=(P0+2σR0)(R0R)3κ(13κR˙cL)2σR4μLR˙RP0PA(t).
      (4)


      For an exhaustive review of the derivation of the different RP-type equations, see
      • Brenner M.P.
      • Hilgenfeldt S.
      • Lohse D.
      Single-bubble sonoluminescence.
      .

      Modeling coated-bubble dynamics

      Adding the interfacial pressure terms

      The gas core of UCA microbubbles is typically coated with a layer of proteins, polymers or phospholipids (
      • Stride E.
      • Saffari N.
      Microbubble ultrasound contrast agents: A review.
      ;
      • Faez T.
      • Emmer M.
      • Kooiman K.
      • Versluis M.
      • van der Steen A.F.W.
      • de Jong N.
      20 years of ultrasound contrast agent modeling.
      ). The first class of agents comprised microbubbles with a rigid shell, often referred to as hard-shelled agents, which motivated the initial approach to modeling their response to ultrasound. This work was started by de Jong and Hoff, who proposed a model describing the behavior of microbubbles of Albunex contrast agent, where the shell was considered to be perfectly elastic (i.e., obeying Hooke's law with no additional damping [
      • de Jong N.
      • Hoff L.
      • Skotland T.
      • Bom N.
      Absorption and scatter of encapsulated gas filled microspheres: Theoretical considerations and some measurements.
      ]). The resulting stiffness pressure term was added in an ad hoc way to the RP equation to describe the bubble dynamics. They later added a semi-empirical shell dissipation term to fit the experimental observations (
      • de Jong N.
      • Hoff L.
      Ultrasound scattering properties of Albunex microspheres.
      ).
      • Church C.C.
      The effect of an elastic solid surface layer on the radial pulsation of gas bubbles.
      and
      • Hoff L.
      • Sontum P.C.
      • Hovem J.M.
      Oscillations of polymeric microbubbles: Effect of the encapsulating shell.
      then followed a similar approach using a more thorough derivation to account for the viscoelastic shell of Albunex. They considered the shell material to be incompressible, but the shear viscosity of the shell was directly included in the derivation of the modified RP equation. Several other models were developed based on linear elasticity (some of them are compared by
      • Sarkar K.
      • Shi W.T.
      • Chatterjee D.
      • Forsberg F.
      Characterization of ultrasound contrast microbubbles using in vitro experiments and viscous and viscoelastic interface models for encapsulation.
      ), presenting similar results. Non-linear elasticity considerations were added by
      • Allen J.S.
      • Rashid M.M.
      Dynamics of a hyperelastic gas-filled spherical shell in a viscous fluid.
      , who considered a non-linear Hooke's law, or neo-Hookean model, for the shell. However, all these models missed the most important aspects of the behavior of the rigid shells, namely (non-spherical) buckling upon compression and rupture upon large expansion. These aspects were investigated and implemented in a model by
      • Marmottant P.
      • Bouakaz A.
      • de Jong N.
      • Quilliet C.
      Buckling resistance of solid shell bubbles under ultrasound.
      . Despite its simple approach, this model captures many features observed in experiments (
      • Lensen D.
      • Gelderblom E.C.
      • Vriezema D.M.
      • Marmottant P.
      • Verdonschot N.
      • Versluis M.
      • de Jong N.
      • van Hest J.C.M.
      Biodegradable polymeric microcapsules for selective ultrasound-triggered drug release.
      ). The model shows, in particular, that these bubbles mostly behave as a free gas bubble after rupture of the shell, as demonstrated before by
      • Postema M.
      • Bouakaz A.
      • Versluis M.
      • de Jong N.
      Ultrasound-induced gas release from contrast agent microbubbles.
      and
      • Bouakaz A.
      • Versluis M.
      • de Jong N.
      High-speed optical observations of contrast agent destruction.
      . A more extensive review of the hard-shelled models is given by
      • Doinikov A.A.
      • Bouakaz A.
      Review of shell models for contrast agent microbubbles.
      .
      To date, the majority of the clinically approved UCAs and research-grade UCAs consist of microbubbles coated with a phospholipid monolayer. In the following, we therefore focus on the modeling of non-linear lipid-coated microbubble dynamics—e.g., the observed driving-pressure–dependent resonance behavior (
      • Chen Q.
      • Zagzebski J.
      • Wilson T.
      • Stiles T.
      Pressure-dependent attenuation in ultrasound contrast agents.
      ;
      • Emmer M.
      • Vos H.J.
      • Goertz D.E.
      • van Wamel A.
      • Versluis M.
      • de Jong N.
      Pressure-dependent attenuation and scattering of phospholipid-coated microbubbles at low acoustic pressures.
      ;
      • Segers T.
      • de Jong N.
      • Versluis M.
      Uniform scattering and attenuation of acoustically sorted ultrasound contrast agents: Modeling and experiments.
      ) and asymmetric oscillation amplitudes—with more pronounced compression than expansion, termed compression-only behavior (
      • de Jong N.
      • Emmer M.
      • Chin C.T.
      • Bouakaz A.
      • Mastik F.
      • Lohse D.
      • Versluis M.
      “Compression-only” behavior of phospholipid-coated contrast bubbles.
      ;
      • Sijl J.
      • Overvelde M.
      • Dollet B.
      • Garbin V.
      • de Jong N.
      • Lohse D.
      • Versluis M.
      “Compression-only” behavior: A second-order nonlinear response of ultrasound contrast agent microbubbles.
      ). Figure 2 (right) gives a schematic of the coated bubble. The viscoelastic lipid coating changes the pressure jump condition across the bubble interface that is used in the RP equation, eqns (1) and (3). The seminal articles by
      • de Jong N.
      • Hoff L.
      • Skotland T.
      • Bom N.
      Absorption and scatter of encapsulated gas filled microspheres: Theoretical considerations and some measurements.
      ,
      • de Jong N.
      • Cornet R.
      • Lancée C.T.
      Higher harmonics of vibrating gas-filled microspheres, part one: Simulations.
      and
      • Church C.C.
      The effect of an elastic solid surface layer on the radial pulsation of gas bubbles.
      include the presence of a shell through the addition of two independent pressure terms on the right-hand side of eqn (4), one accounting for the elasticity Pelas of the shell and one accounting for the viscous dissipation Pvis in the shell. These terms can be derived for a shell of finite thickness (
      • Church C.C.
      The effect of an elastic solid surface layer on the radial pulsation of gas bubbles.
      ;
      • Hoff L.
      • Sontum P.C.
      • Hovem J.M.
      Oscillations of polymeric microbubbles: Effect of the encapsulating shell.
      ) and for an infinitesimally thin surface layer using material constitutive laws (
      • de Jong N.
      • Hoff L.
      • Skotland T.
      • Bom N.
      Absorption and scatter of encapsulated gas filled microspheres: Theoretical considerations and some measurements.
      ;
      • de Jong N.
      • Hoff L.
      Ultrasound scattering properties of Albunex microspheres.
      ;
      • de Jong N.
      • Cornet R.
      • Lancée C.T.
      Higher harmonics of vibrating gas-filled microspheres, part one: Simulations.
      ;
      • Chatterjee D.
      • Sarkar K.
      A Newtonian rheological model for the interface of microbubble contrast agents.
      ;
      • Marmottant P.
      • van der Meer S.
      • Emmer M.
      • Versluis M.
      • de Jong N.
      • Hilgenfeldt S.
      • Lohse D.
      A model for large amplitude oscillations of coated bubbles accounting for buckling and rupture.
      ;
      • Sarkar K.
      • Shi W.T.
      • Chatterjee D.
      • Forsberg F.
      Characterization of ultrasound contrast microbubbles using in vitro experiments and viscous and viscoelastic interface models for encapsulation.
      ;
      • Tsiglifis K.
      • Pelekasis N.A.
      Nonlinear radial oscillations of encapsulated microbubbles subject to ultrasound: The effect of membrane constitutive law.
      ;
      • Doinikov A.A.
      • Haac J.F.
      • Dayton P.A.
      Modeling of nonlinear viscous stress in encapsulating shells of lipid-coated contrast agent microbubbles.
      ). The physical principles of the available models for soft-shelled models have been exhaustively reviewed by
      • Doinikov A.A.
      • Bouakaz A.
      Review of shell models for contrast agent microbubbles.
      . For a more extensive review, see
      • Faez T.
      • Emmer M.
      • Kooiman K.
      • Versluis M.
      • van der Steen A.F.W.
      • de Jong N.
      20 years of ultrasound contrast agent modeling.
      .
      The viscous pressure term Pvis is added to account for viscous energy dissipation in the microbubble shell resulting from the work done to change the distance between the lipid molecules during an acoustic cycle. The most common form of Pvis is derived from Newton's linear viscous law for an infinitesimally thin shell (
      • de Jong N.
      • Cornet R.
      • Lancée C.T.
      Higher harmonics of vibrating gas-filled microspheres, part one: Simulations.
      ;
      • Church C.C.
      The effect of an elastic solid surface layer on the radial pulsation of gas bubbles.
      ;
      • Morgan K.
      • Allen J.
      • Dayton P.
      • Chomas J.
      • Klibanov A.
      • Ferrara K.
      Experimental and theoretical evaluation of microbubble behavior: Effect of transmitted phase and bubble size.
      ;
      • Chatterjee D.
      • Sarkar K.
      A Newtonian rheological model for the interface of microbubble contrast agents.
      ;
      • Marmottant P.
      • van der Meer S.
      • Emmer M.
      • Versluis M.
      • de Jong N.
      • Hilgenfeldt S.
      • Lohse D.
      A model for large amplitude oscillations of coated bubbles accounting for buckling and rupture.
      ;
      • Sarkar K.
      • Shi W.T.
      • Chatterjee D.
      • Forsberg F.
      Characterization of ultrasound contrast microbubbles using in vitro experiments and viscous and viscoelastic interface models for encapsulation.
      ):
      Pvis=4κsR˙R2,
      (5)


      with κs the surface dilatational viscosity of the lipid monolayer shell. Typically, it is assumed that κs does not depend on the bubble surface area, surface dilatation, or surface dilatation rate. The validity of eqn (5) will be discussed in the final paragraph of this section.
      The elastic pressure term Pelas accounts for the pressure increase inside the bubble owing to the stiffness of the shell. Shell elasticity χ can be described to result from interfacial tension gradients induced by (acoustically-driven) bubble surface-area variations dA (
      • Glazman R.E.
      Effects of adsorbed films on gas bubble radial oscillations.
      ;
      • Mohwald H.
      Handbook of biological physics.
      ;
      • Marmottant P.
      • van der Meer S.
      • Emmer M.
      • Versluis M.
      • de Jong N.
      • Hilgenfeldt S.
      • Lohse D.
      A model for large amplitude oscillations of coated bubbles accounting for buckling and rupture.
      ;
      • Sarkar K.
      • Shi W.T.
      • Chatterjee D.
      • Forsberg F.
      Characterization of ultrasound contrast microbubbles using in vitro experiments and viscous and viscoelastic interface models for encapsulation.
      ):
      χ=AdσdA.
      (6)


      Assuming a constant and surface-area–independent shell elasticity, eqn (6) can be integrated to find the effective interfacial tension: σ(R)=χ(R2/R021), with R0 the equilibrium bubble radius. With the effective interfacial tension, the elastic pressure term reads
      Pelas=2σ(R)R=2χR(R2R021).
      (7)


      Equation (7) is the most common form of the elastic pressure term (
      • de Jong N.
      • Cornet R.
      • Lancée C.T.
      Higher harmonics of vibrating gas-filled microspheres, part one: Simulations.
      ;
      • Church C.C.
      The effect of an elastic solid surface layer on the radial pulsation of gas bubbles.
      ;
      • Chatterjee D.
      • Sarkar K.
      A Newtonian rheological model for the interface of microbubble contrast agents.
      ;
      • Marmottant P.
      • van der Meer S.
      • Emmer M.
      • Versluis M.
      • de Jong N.
      • Hilgenfeldt S.
      • Lohse D.
      A model for large amplitude oscillations of coated bubbles accounting for buckling and rupture.
      ;
      • Sarkar K.
      • Shi W.T.
      • Chatterjee D.
      • Forsberg F.
      Characterization of ultrasound contrast microbubbles using in vitro experiments and viscous and viscoelastic interface models for encapsulation.
      ;
      • Tsiglifis K.
      • Pelekasis N.A.
      Nonlinear radial oscillations of encapsulated microbubbles subject to ultrasound: The effect of membrane constitutive law.
      ). The full equation—including the shell pressure terms and eqns (5) and (7)—then reads:
      ρL(RR¨+32R˙2)=(P0+2σ(R0)R0)(R0R)3κ(13κR˙cL)P0PA(t)4μLR˙R2χR(R2R021)4κsR˙R2,
      (8)


      where σ(R0) represents the surface tension of the bubble at rest.
      With a complete description of the radial oscillations of coated bubbles, we can now calculate the echo produced by UCA microbubbles. The sound emitted by an oscillating microbubble has two contributions. The first is a passive contribution that simply results from the mismatch in acoustic impedance between the gas core and the surrounding medium. The second one is the active part and results from the volumetric oscillations V(t) of the bubble. Micrometer-sized bubbles driven near resonance (see also the next section) have a size that is typically three orders of magnitude smaller than the incident wavelength. It has been shown that in this case the passive contribution is negligible with respect to the active part (
      • Hilgenfeldt S.
      • Lohse D.
      • Zomack M.
      Response of bubbles to diagnostic ultrasound: A unifying theoretical approach.
      ). From mass and momentum conservation it follows that the active scattered pressure wave Ps at a distance r from the bubble center equals (
      • Vokurka K.
      On Rayleigh's model of a freely oscillating bubble. 1: Basic relations.
      ):
      Ps=ρL4πrV¨=ρLr(R2R¨+2RR˙2).
      (9)


      Thus, the scattered pressure Ps(t) can be directly calculated by inserting into eqn (9) the solution of the RP equation, R(t), and its derivatives.
      The numerical solution R(t) of the ordinary differential equation (ODE) given by eqn (8) is displayed in Figure 3 (left) for a bubble with radius 2.0 μm. Three driving pressures were used here—5, 50 and 500 kPa—corresponding to bubble oscillation behavior in the linear, non-linear and inertial-collapse regimes, respectively. The scattered pressure Ps(t) was calculated by inserting R(t) and its derivatives R˙ and R¨ directly into eqn (9) (see Fig. 3, middle). Figure 3 (right) gives the spectral content of the scattered pressure. It nicely shows the different regimes of bubble oscillation behavior: in the linear regime, the response of the bubble is at the fundamental frequency of 2.0 MHz; for a driving pressure of 50 kPa the response is non-linear, with buildup of second harmonic and superharmonic components and some small amount of broadband emission; and for the highest-pressure regime, despite some harmonic content, the response is primarily broadband owing to the impulsive collapses of the bubble.
      Fig 3
      Fig. 3(Left) Radius-time R(t) curves of a bubble with a radius R0 = 2.0 μm at three different driving pressures: PA = 5, 50 and 500 kPa. The parameters for this system are as follows: ambient pressure P0 = 100 kPa; surface tension σ = 72 mN/m; polytropic exponent κ = 1.07; speed of sound cL = 1500 m/s; liquid density ρL = 1000 kg/m3; liquid viscosity μL = 2.0 mPa s (to account for thermal damping); shell elasticity χ = 1.0 N/m; shell viscosity κs = 1.0 × 10−8 kg/s; and initial surface tension σ(R0) = 20 mN/m. Ultrasound consisted of a Gaussian-tapered narrowband driving pulse with 16 cycles at a driving frequency of 2.0 MHz. (Middle) Scattered pressure Ps(t) at a distance of r = 2.54 cm at the three driving pressures. (Right) Spectral content of the scattered pressures.
      Figure 4a shows the maximum bubble response, expressed as the normalized difference of maximum and minimum bubble amplitude, as a function of the bubble radius for a driving frequency of 1.0 MHz and a driving pressure of 75 kPa. Bubble shell and liquid properties are the same as in Figure 3. The resonance behavior is clearly visible, with bubbles with a radius of 3.2 μm displaying maximum oscillation amplitudes. Figure 4B shows the fundamental and second harmonic components of the scattered response of the bubbles driven in Figure 4a. It shows first of all that next to the resonance behavior, bubbles of a larger size also contribute to the scattering at the fundamental driving frequency. More importantly, it shows that the non-linear generation of the second harmonic is limited to the resonant bubbles.
      Fig 4
      Fig. 4(a) Maximum bubble response, expressed as the normalized amplitude (i.e., difference of maximum and minimum bubble amplitudes normalized by 2R0) as a function of the bubble radius for a driving frequency of 1.0 MHz and a driving pressure of 75 kPa. Other parameters the same as in . (b) The fundamental and second harmonic components of the scattered response of the bubble driven in (a) at a distance r = 2.54 cm. (c) The total damping δtot of an uncoated bubble in water as a function of bubble size with damping contributions from sound reradiation δrad, viscous damping δvis owing to liquid viscosity and thermal damping δth. (d) The total damping of a coated bubble (χ = 0.5 N/m) over the size range relevant for ultrasound contrast agents. The shell damping δshell was calculated using a fixed shell viscosity κs = 1.0 × 10−8 kg/s. Note that the total damping of a coated bubble is dominated by shell damping for bubbles in the diameter range 2–3 μm.

      Linear bubble-shell modeling: Resonance frequency and damping

      The role of the shell in the microbubble dynamics becomes clear through linearization of the RP equation (eqn [8]), by assuming sinusoidal acoustic driving and small amplitudes of oscillation with respect to the equilibrium radius (i.e., R=R0(1+ε(t)), with  ε (t) ≪ 1), and by neglecting higher-order terms to yield the classic equation of a driven harmonic oscillator:
      ε¨+δtotω0ε˙+ω02x=PAρR0sinωt,
      (10)


      with ε˙ and ε¨ the first- and second-order derivatives of  ε  with respect to time, δtot the total damping of the system, ω the angular ultrasound frequency and ω0 the angular eigenfrequency of the oscillator:
      ω0=2πf0=1R01ρL(3κP0+(3κ1)2σ(R0)R0+4χR0).
      (11)


      In these equations, σ(R0) represents the surface tension of the bubble at rest. Equation (11) shows that the eigenfrequency of a bubble is inversely proportional to the microbubble size and that the stiffness χ of the coating increases the eigenfrequency of the coated bubble compared with the free gas bubble.
      The total damping δtot is a summation of the damping contributions that arise from reradiation of sound, viscous dissipation in the surrounding liquid and shell viscous dissipation: δtot=δrad+δvis+δshell. The linearization of eqn (8) results in the following expressions for the damping terms at ω = ω0:
      δrad=3κP0ρLω0cLR0,δvis=4μLρLω0R02,δshell=4κsρLω0R03.
      (12)


      This expression for radiation damping is in approximate agreement with the exact form δrad=ωR0/cL (
      • Leighton T.G.
      The acoustic bubble.
      ). A microbubble also experiences thermal damping δth owing to heat diffusion (
      • Prosperetti A.
      Thermal effects and damping mechanisms in the forced radial oscillations of gas bubbles in liquids.
      ;
      • Leighton T.G.
      The acoustic bubble.
      ). For micrometer-sized bubbles it is of the same order as the viscous damping for a bubble in water (
      • Devin Jr, C.
      Survey of thermal, radiation, and viscous damping of pulsating air bubbles in water.
      ), and therefore typically the viscosity of the medium is increased phenomenologically by a factor of 2 to account for thermal damping (
      • van der Meer S.
      • Dollet B.
      • Voormolen M.
      • Chin C.T.
      • Bouakaz A.
      • de Jong N.
      • Versluis M.
      • Lohse D.
      Microbubble spectroscopy of ultrasound contrast agents.
      ;
      • Sijl J.
      • Dollet B.
      • Overvelde M.
      • Garbin V.
      • Rozendal T.
      • de Jong N.
      • Lohse D.
      • Versluis M.
      Subharmonic behavior of phospholipid-coated ultrasound contrast agent microbubbles.
      ;
      • Segers T.
      • de Jong N.
      • Versluis M.
      Uniform scattering and attenuation of acoustically sorted ultrasound contrast agents: Modeling and experiments.
      ). The damping constants for an uncoated bubble are plotted in Figure 4c. Figure 4d shows the damping constants for a coated bubble (χ = 0.5 N/m). Shell damping was calculated using a fixed shell viscosity κs = 1.0 × 10−8 kg/s. Note that the damping for a coated bubble is dominated by the shell viscous damping, as it contributes approximately 75% of the total damping for bubbles in the relevant size range of 2–3 μm. The total damping δtot of the system results in a slightly reduced microbubble resonance frequency fres as compared with its eigenfrequency f0: fres=f01δtot2/2. For a typical total damping of 0.25, however, the decrease in resonance frequency is less than 2%. On the other hand, eqn (11) demonstrates that the microbubble shell stiffness can dramatically increase the resonance frequency. Figure 5 compares the resonance curve of a free gas bubble (R0 = 2.5 μm) with that of lipid-coated bubbles of the same sizes, with respective shell stiffness of 1.0 and 2.5 N/m and shell viscosities of κs = 6 × 10−9 and 12 × 10−9 kg/s. Note that the linearization performed here provides insightful information only for coated microbubbles driven at very low pressure amplitudes (<5 kPa).
      Fig 5
      Fig. 5Amplitude of oscillation normalized to that of the uncoated bubble (R0 = 2.5 μm). The resonance curves shift toward higher frequencies with increasing shell stiffness χ and broaden with increasing shell damping κs. The other parameters are as follows: driving pressure PA = 1 kPa; ambient pressure P0 = 100 kPa; surface tension of water σ = 72 mN/m; polytropic exponent κ = 1.07; liquid density ρL = 1000 kg/m3; speed of sound in the liquid cL = 1500 m/s; liquid viscosity μL = 2 mPa s (to also account for thermal damping); and σ(R0) = 20 mN/m.

      Non-linear bubble-shell modeling: Buckling and rupture

      For elevated acoustic pressures we will need to describe the non-linear dynamics of the coated bubble, which is then modeled by solving the full RP equation including the shell pressure terms (eqn [8]). The RP equation is non-linear owing to the intrinsic non-linearities in the ODE. However, the shell pressure terms are linear (i.e., the surface tension is linearly dependent on the microbubble surface area) and can adopt negative values or values exceeding the surface tension of the surrounding medium (see the black dotted line in Fig. 6a), and the shell stiffness χ is constant throughout and independent of the surface dilatation (see Fig. 6b). Therefore, eqn 8 does not predict the experimentally observed non-linear microbubble dynamics such as compression-only behavior (
      • de Jong N.
      • Emmer M.
      • Chin C.T.
      • Bouakaz A.
      • Mastik F.
      • Lohse D.
      • Versluis M.
      “Compression-only” behavior of phospholipid-coated contrast bubbles.
      ) and the resonance-frequency shift with increasing acoustic pressure amplitudes for lipid-coated bubbles (
      • Overvelde M.
      • Garbin V.
      • Sijl J.
      • Dollet B.
      • de Jong N.
      • Lohse D.
      • Versluis M.
      Nonlinear shell behavior of phospholipid-coated microbubbles.
      ).
      Fig 6
      Fig. 6(a) Ad hoc surface-tension curves implemented in Rayleigh–Plesset–type equations to account for shell elasticity (reproduced from
      • Segers T.
      • Gaud E.
      • Versluis M.
      • Frinking P.
      High-precision acoustic measurements of the nonlinear dilatational elasticity of phospholipid coated monodisperse microbubbles.
      by permission of The Royal Society of Chemistry); (b) shell stiffness (χ = Adσ / dA) curves corresponding to the surface-tension curves plotted in (a) (reproduced from
      • Segers T.
      • Gaud E.
      • Versluis M.
      • Frinking P.
      High-precision acoustic measurements of the nonlinear dilatational elasticity of phospholipid coated monodisperse microbubbles.
      by permission of The Royal Society of Chemistry); (c) surface tension as a function of area per lipid molecule measured at slow dilatational rates in a Langmuir trough for disaturated phosphatidylcholine (DSPC) (solid line) and dipalmitoylphosphatidylglycerol (DPPG) (dotted line; reproduced with permission from
      • Marmottant P.
      • van der Meer S.
      • Emmer M.
      • Versluis M.
      • de Jong N.
      • Hilgenfeldt S.
      • Lohse D.
      A model for large amplitude oscillations of coated bubbles accounting for buckling and rupture.
      ); (d) “compression-only” (red curve) and “expansion-only” (black curve) behavior modeled by limiting the surface tension between zero and that of the surrounding medium (solid curves in a); (e) measured surface-tension curve compared with Marmottant's ad hoc implementation (reproduced from
      • Segers T.
      • Gaud E.
      • Versluis M.
      • Frinking P.
      High-precision acoustic measurements of the nonlinear dilatational elasticity of phospholipid coated monodisperse microbubbles.
      by permission of The Royal Society of Chemistry); (f) measured non-linear dilatational elasticity as a function of surface area (reproduced from
      • Segers T.
      • Gaud E.
      • Versluis M.
      • Frinking P.
      High-precision acoustic measurements of the nonlinear dilatational elasticity of phospholipid coated monodisperse microbubbles.
      by permission of The Royal Society of Chemistry).
      The breakthrough in modeling non-linear bubble dynamics came from
      • Marmottant P.
      • van der Meer S.
      • Emmer M.
      • Versluis M.
      • de Jong N.
      • Hilgenfeldt S.
      • Lohse D.
      A model for large amplitude oscillations of coated bubbles accounting for buckling and rupture.
      . Inspired by Langmuir-trough surface pressure-area isotherms of lipid monolayers (Fig. 6c), they limited the surface tension to zero for a compression below a certain buckling radius Rb and to the surface tension of the surrounding medium for an expansion above a so-called rupture radius Rr:
      σ(R)={0ifRRbχ(R2Rb21)ifRbRRrσwaterifRRr.
      (13)


      The main idea is that during the compression phase, the lipid monolayer cannot withstand compression beyond the buckling radius, at which the shell buckles, resulting in a tensionless shell (χ=Adσ/dA=0); see Figure 1j, where buckling is captured by ultrahigh-speed imaging. During the expansion phase, the average area per lipid molecule increases, either owing to shell rupture (e.g., along lipid domains [
      • Borden M.A.
      • Martinez G.V.
      • Ricker J.
      • Tsvetkova N.
      • Longo M.
      • Gillies R.J.
      • Dayton P.A.
      • Ferrara K.W.
      Lateral phase separation in lipid-coated microbubbles.
      ]) or owing to a more uniform increase in the area per lipid molecule, resulting in an increase of the effective surface tension until the surface tension σwater of the surrounding medium is reached, again leading to a tensionless state (χ=Adσ/dA=0). In the elastic phase, the surface tension was assumed to increase monotonically with surface area, as in eqn (7). The solid black line in Figure 6a shows the surface-tension curve used in the Marmottant model. Note that purely elastic (linear) oscillations are limited to small amplitudes of oscillation around the initial bubble surface area with radius R0. Figure 6b shows the corresponding elasticity χ as a function of the surface area.
      When the bubble is driven slightly outside the elastic regime, there are three possible oscillation scenarios, depending on the initial surface tension σ(R0) of the bubble. At σ(R0) < σwater/2, the bubble can reach the buckling state during compression while it remains in the elastic state during expansion. During the buckling phase, the shell stiffness vanishes and the bubble therefore experiences less resistance against compression than against expansion. This asymmetry in shell stiffness over an oscillation period induces compression-only behavior, as shown in Figure 6d. Similarly, at σ(R0) > σwater/2, the bubble experiences less resistance upon expansion than during compression once it is driven beyond the rupture radius. This drives the asymmetric expansion-only behavior (Fig. 6d). At σ(R0)=σwater/2, the bubble oscillations are symmetric.
      The non-linear interfacial tension and corresponding shell stiffness have another implication for a coated bubble driven outside its elastic regime, namely a downshift of its resonance frequency. The downshift results from the effective averaging of the shell stiffness over elastic and non-elastic regimes. When the acoustic pressure is increased, the effective shell stiffness decreases. At high driving pressures, the contribution of shell stiffness becomes negligible and the resonance frequency approaches that of an uncoated bubble. This sudden increase in microbubble response above an acoustic pressure threshold has also been named thresholding behavior (
      • Emmer M.
      • van Wamel A.
      • Goertz D.E.
      • de Jong N.
      The onset of microbubble vibration.
      ). A detailed explanation of thresholding behavior is given by
      • Overvelde M.
      • Garbin V.
      • Sijl J.
      • Dollet B.
      • de Jong N.
      • Lohse D.
      • Versluis M.
      Nonlinear shell behavior of phospholipid-coated microbubbles.
      .
      Next to compression-only and driving-pressure–dependent resonance behavior, the Marmottant model has also proven successful in modeling the generation of subharmonics (
      • Frinking P.J.A.
      • Gaud E.
      • Brochot J.
      • Arditi M.
      Subharmonic scattering of phospholipid-shell microbubbles at low acoustic pressure amplitudes.
      ;
      • Sijl J.
      • Dollet B.
      • Overvelde M.
      • Garbin V.
      • Rozendal T.
      • de Jong N.
      • Lohse D.
      • Versluis M.
      Subharmonic behavior of phospholipid-coated ultrasound contrast agent microbubbles.
      ). It has been shown that subharmonics are induced by rapid change in shell stiffness with a small change in surface area (
      • Sijl J.
      • Dollet B.
      • Overvelde M.
      • Garbin V.
      • Rozendal T.
      • de Jong N.
      • Lohse D.
      • Versluis M.
      Subharmonic behavior of phospholipid-coated ultrasound contrast agent microbubbles.
      ,
      • Sijl J.
      • Overvelde M.
      • Dollet B.
      • Garbin V.
      • de Jong N.
      • Lohse D.
      • Versluis M.
      “Compression-only” behavior: A second-order nonlinear response of ultrasound contrast agent microbubbles.
      ). Thus, the second-order derivative of the interfacial tension curve with respect to the bubble surface area is the driving factor behind the generation of subharmonics. However,
      • Marmottant P.
      • van der Meer S.
      • Emmer M.
      • Versluis M.
      • de Jong N.
      • Hilgenfeldt S.
      • Lohse D.
      A model for large amplitude oscillations of coated bubbles accounting for buckling and rupture.
      implemented the dilatational interfacial tension and corresponding shell stiffness in a rather unphysical way, as they are discontinuous at the buckling and rupture radii (Fig. 6a, 6b). Therefore, the numerical solutions of eqns (8) and (13) are sensitive to the integration time step of the ODE solver, as this sets d2σ/dA2. To circumvent this problem,
      • Sijl J.
      • Dollet B.
      • Overvelde M.
      • Garbin V.
      • Rozendal T.
      • de Jong N.
      • Lohse D.
      • Versluis M.
      Subharmonic behavior of phospholipid-coated ultrasound contrast agent microbubbles.
      proposed extending the Marmottant model with two crossover functions for the transition from the regions with zero elasticity to the one with finite elasticity (see the blue curves in Fig. 6a, 6b). With the aim of getting closer to a more physicochemical surface-area–dependent surface tension (i.e., with decreasing elasticity for increasing area per lipid molecule)
      • Paul S.
      • Katiyar A.
      • Sarkar K.
      Material characterization of the encapsulation of an ultrasound contrast microbubble and its subharmonic response: Strain-softening interfacial elasticity model.
      tested a quadratic elasticity model and an exponential elasticity model (see Fig. 6a, 6b). They once again pointed out the importance of a surface-area–dependent elasticity for non-linear bubble dynamics.
      It has now become clear that the actual shape of the surface-tension curve and the initial surface tension are important for the correct prediction of non-linear radial bubble dynamics. In the modeling efforts, ad hoc assumptions have been made regarding the rheological properties of the microbubble shell, as measurement of the actual σ(R) dependence of a lipid-coated microbubble interface is difficult at megahertz frequencies (
      • Marmottant P.
      • van der Meer S.
      • Emmer M.
      • Versluis M.
      • de Jong N.
      • Hilgenfeldt S.
      • Lohse D.
      A model for large amplitude oscillations of coated bubbles accounting for buckling and rupture.
      ;
      • Stride E.
      The influence of surface adsorption on microbubble dynamics.
      ). The difficulty lies in the extremely short timescales and small length scales of ultrasound-driven microbubble oscillations, which render unuseful the relatively slow measurements obtained by Langmuir trough (
      • Lozano M.M.
      • Longo M.L.
      Complex formation and other phase transformations mapped in saturated phosphatidylcholine/DSPE-PEG2000 monolayers.
      ) or bubble tensiometer (
      • Lee S.
      • Kim D.H.
      • Needham D.
      Equilibrium and dynamic interfacial tension measurements at microscopic interfaces using a micropipet technique. 2: Dynamics of phospholipid monolayer formation and equilibrium tensions at the water-air interface.
      ). Furthermore, UCA bubbles are at least one order of magnitude smaller than the minimum cap radius in a bubble tensiometer (
      • Alvarez N.J.
      • Walker L.M.
      • Anna S.L.
      A microtensiometer to probe the effect of radius of curvature on surfactant transport to a spherical interface.
      ), and the curvature of a monomolecular film may play a role in its viscoelastic properties (
      • Kotula A.P.
      • Anna S.L.
      Insoluble layer deposition and dilatational rheology at a microscale spherical cap interface.
      ). Recently, the dilatational shell elasticity curve of lipid-coated bubbles has been measured experimentally by controlling the surface area of monodisperse bubbles through the ambient pressure and measuring the resonance behavior from acoustic attenuation measurements (
      • Segers T.
      • Gaud E.
      • Versluis M.
      • Frinking P.
      High-precision acoustic measurements of the nonlinear dilatational elasticity of phospholipid coated monodisperse microbubbles.
      ). The measured χ(A/A0) curve, Figure 6f, has been integrated to find the σ(A/A0) curve, Figure 6e. The measured dilatational interfacial tension curve shows remarkable agreement with the initial assumptions of (Marmottant 2005), explaining the effectiveness of their model in the prediction of non-linear bubble dynamics. Is has also been shown that the χ(A/A0) curve is independent of the total bubble surface area for a given lipid composition (
      • Segers T.
      • Gaud E.
      • Versluis M.
      • Frinking P.
      High-precision acoustic measurements of the nonlinear dilatational elasticity of phospholipid coated monodisperse microbubbles.
      ); in other words, it is independent of the bubble size. However, it is also shown that the shell stiffness is not constant in the elastic regime (see Fig. 6f), which may result in a change in acoustic properties of a dilute bubble suspension over time through gas loss.

      Modeling shell viscosity

      In contrast to the bubble-size–independent shell stiffness, the reported shell viscosity values are strongly dependent on the equilibrium bubble size. Shell viscosity κs has been extensively characterized through a direct fit of eqn (8) to experimental R(t) curves, through optical light scattering (
      • Tu J.
      • Guan J.
      • Qiu Y.
      • Matula T.J.
      Estimating the shell parameters of SonoVue microbubbles using light scattering.
      ) and through acoustic and high-speed imaging characterization (
      • Morgan K.
      • Allen J.
      • Dayton P.
      • Chomas J.
      • Klibanov A.
      • Ferrara K.
      Experimental and theoretical evaluation of microbubble behavior: Effect of transmitted phase and bubble size.
      ;
      • van der Meer S.
      • Dollet B.
      • Voormolen M.
      • Chin C.T.
      • Bouakaz A.
      • de Jong N.
      • Versluis M.
      • Lohse D.
      Microbubble spectroscopy of ultrasound contrast agents.
      ;
      • Overvelde M.
      • Garbin V.
      • Sijl J.
      • Dollet B.
      • de Jong N.
      • Lohse D.
      • Versluis M.
      Nonlinear shell behavior of phospholipid-coated microbubbles.
      ;
      • Segers T.
      • Gaud E.
      • Versluis M.
      • Frinking P.
      High-precision acoustic measurements of the nonlinear dilatational elasticity of phospholipid coated monodisperse microbubbles.
      ). The data show an exponential increase of κs with bubble size, which indicates an incomplete implementation of shell viscous dissipation in eqn (5). Shell viscosity has been shown to decrease with dilatation rate (
      • van der Meer S.
      • Dollet B.
      • Voormolen M.
      • Chin C.T.
      • Bouakaz A.
      • de Jong N.
      • Versluis M.
      • Lohse D.
      Microbubble spectroscopy of ultrasound contrast agents.
      ), which indicates shear-thinning behavior. Based on those experimental data,
      • Doinikov A.A.
      • Haac J.F.
      • Dayton P.A.
      Modeling of nonlinear viscous stress in encapsulating shells of lipid-coated contrast agent microbubbles.
      included shear-thinning behavior in the RP equation in an empirical way. Furthermore,
      • Segers T.
      • Gaud E.
      • Versluis M.
      • Frinking P.
      High-precision acoustic measurements of the nonlinear dilatational elasticity of phospholipid coated monodisperse microbubbles.
      have measured a dependence of κs on the bubble surface area (lipid-packing density), indicating that not only the dilatation rate is important but also surface dilatation itself. The future challenge is to clarify this matter, including the role of different shell components on the acoustic bubble response. Attempts have been made to model shell parameters based on molecular interactions (
      • Stride E.
      The influence of surface adsorption on microbubble dynamics.
      ;
      • Borden M.A.
      Intermolecular forces model for lipid microbubble shells.
      ). This route can be further explored, for example using molecular-dynamics simulations (
      • Boek E.S.
      • Padding J.T.
      • den Otter W.K.
      • Briels W.J.
      Mechanical properties of surfactant bilayer membranes from atomistic and coarse-grained molecular dynamics simulations.
      ;
      • van Opheusden J.H.J.
      • Molenaar J.
      Lipid aggregate formation at an oscillating bubble surface: A simulation study.
      ). Control over the bubble-size distribution has now been achieved (Fig. 1g;
      • Segers T.
      • de Rond L.
      • de Jong N.
      • Borden M.
      • Versluis M.
      Stability of monodisperse phospholipid-coated microbubbles formed by flow-focusing at high production rates.
      ,
      • Segers T.
      • Lohse D.
      • Versluis M.
      • Frinking P.
      Universal equations for the coalescence probability and long-term size stability of phospholipid-coated monodisperse microbubbles formed by flow-focusing.
      ,
      • Segers T.
      • Lassus A.
      • Bussat P.
      • Gaud E.
      • Frinking P.
      Improved coalescence stability of monodisperse phospholipid-coated microbubbles formed by flow-focusing at elevated temperatures.
      ). Together with monodispersity, control over the acoustic response of coated bubbles would be highly valuable, as this would allow the tailoring of microbubble design for specific clinical applications, e.g. non-invasive blood-pressure measurements, and therapeutic applications [
      • Miller D.L.
      • Lu X.
      • Dou C.
      • Fabiilli M.L.
      • Church C.C.
      The dependence of glomerular capillary hemorrhage induced by contrast enhanced diagnostic ultrasound on microbubble diameter.
      ], including sonothrombolysis and drug and gene delivery using bubbles and ultrasound.

      Scattering and attenuation

      Scattering leads to a loss of energy of the insonifying acoustic wave. The acoustic energy that is lost by the scattering of a single bubble, assuming that the scattering is purely spherical, can be expressed as the ratio of the scattered power to the incident acoustic intensity, also known as the scattering cross section σs:
      σs=4πr2|Ps(r)|2|PT|2,
      (14)


      with r the distance from the bubble where the pressure is measured, |Ps(r)| the amplitude of the scattered pressure and |PT| the pressure amplitude of the incident acoustic wave (
      • Hilgenfeldt S.
      • Lohse D.
      • Zomack M.
      Response of bubbles to diagnostic ultrasound: A unifying theoretical approach.
      ). This expression also holds at the bubble wall, at r = R0. The frequency-dependent scattering cross section can be calculated by using the power spectrum of Ps(r) in eqn (14) instead of the total scattered power. Note that through Ps, the scattering cross section is now a function of the shell parameters and depends on the frequency and the pressure amplitude of the acoustic wave.
      For small-amplitude bubble oscillations driven in the linear regime at low acoustic pressure amplitudes, the linearized scattering cross section Ωs can be expressed as follows (
      • de Jong N.
      • Hoff L.
      • Skotland T.
      • Bom N.
      Absorption and scatter of encapsulated gas filled microspheres: Theoretical considerations and some measurements.
      ;
      • Leighton T.G.
      The acoustic bubble.
      ):
      Ωs=4πR02((ω0/ω)21)2+δtot2.
      (15)


      Equation (15) shows that for a bubble driven at resonance (ω=ω0) and with a total damping of 0.25 (which is typical for UCA bubbles), the scattering cross section is two orders of magnitude larger than its geometric cross-sectional area.
      In addition to scattering, energy of the insonifying ultrasound wave is dissipated through viscous damping, thermal damping and shell damping. The total energy loss of the acoustic wave owing to the scattering and damping can therefore be expressed as the ratio of the total energy loss per bubble to the intensity of the acoustic wave: the extinction cross section σe. It is found by multiplying the scattering cross section σs and the ratio of the total damping δtot to the radiation damping δrad (
      • Medwin H.
      Counting bubbles acoustically: A review.
      ).
      The total amount of scattered and absorbed energy of an acoustic wave propagating through a dilute cloud of homogeneously distributed bubbles at concentrationn(R) of bubbles with radius R can now be obtained by summation over the scattering and extinction cross sections of all the individual bubbles present in the acoustic beam (
      • de Jong N.
      • Hoff L.
      • Skotland T.
      • Bom N.
      Absorption and scatter of encapsulated gas filled microspheres: Theoretical considerations and some measurements.
      ;
      • Raymond J.L.
      • Haworth K.J.
      • Bader K.B.
      • Radhakrishnan K.
      • Griffin J.K.
      • Huang S.L.
      • McPherson D.D.
      • Holland C.K.
      Broadband attenuation measurements of phospholipid-shelled ultrasound contrast agents.
      ). This linearized approach only holds if the pressure wave scattered by a neighboring bubble at a distance d is much smaller than the average insonifying pressure field: |Ps(d)|/|PT| ≪ 1. At a bubble concentration n, the average distance between bubbles is d = n−1/3, which, using eqn (14), results in the condition n(R)2/3Ωs ≪ 1 (
      • Commander K.W.
      • Prosperetti A.
      Linear pressure waves in bubbly liquids: Comparison between theory and experiment.
      ;
      • Goertz D.E.
      • de Jong N.
      • van der Steen A.F.W.
      Attenuation and size distribution measurements of Definity and manipulated Definity populations.
      ). The total attenuation α of an ultrasound wave traveling through a bubble suspension is a result of the total energy dissipation resulting from the presence of bubbles. For linear bubble oscillations, it is given by
      α=10ln(10)Rδtotδradn(R)Ωsd,
      (16)


      with d the acoustic path length over which the sound wave interacts with the bubbles. Similarly, again for linear bubble oscillations, the backscatter coefficient η can be calculated as follows (
      • Marsh J.N.
      • Hughes M.S.
      • Brandenburger G.H.
      • Miller J.G.
      Broadband measurement of the scattering-to-attenuation ratio for Albunex at 37°C.
      ;
      • Gorce J.M.
      • Arditi M.
      • Schneider M.
      Influence of bubble size distribution on the echogenicity of ultrasound contrast agents: A study of SonoVue.
      ):
      η=14πRn(R)Ωs.
      (17)


      As discussed before, when the acoustic pressure is increased, the effective shell stiffness decreases. At high driving pressures, the contribution of shell stiffness then becomes negligible and the resonance frequency approaches that of an uncoated bubble. The decrease of the resonance frequency with increasing acoustic pressure is shown in Figure 7a for measured attenuation spectra of a monodisperse microbubble suspension (
      • Segers T.
      • de Jong N.
      • Versluis M.
      Uniform scattering and attenuation of acoustically sorted ultrasound contrast agents: Modeling and experiments.
      ). The shift in resonance frequency can be extremely useful in contrast imaging schemes such as power modulation (
      • Eckersley R.J.
      • Chin C.T.
      • Burns P.N.
      Optimising phase and amplitude modulation schemes for imaging microbubble contrast agents at low acoustic power.
      ). The transmit frequency can be chosen such that it coincides with the microbubble resonance frequency at the higher acoustic pressures, generating a strong echo only when the transmit power exceeds the threshold for the resonance-frequency downshift.
      Fig 7
      Fig. 7(a) Attenuation of a monodisperse bubble sample (monodispersity achieved by acoustic sorting) measured at pressures ranging from 5 to 75 kPa (
      • Segers T.
      • Versluis M.
      Acoustic bubble sorting for ultrasound contrast agent enrichment.
      ); (B) modeled attenuation (reproduced with permission from
      • Segers T.
      • de Jong N.
      • Versluis M.
      Uniform scattering and attenuation of acoustically sorted ultrasound contrast agents: Modeling and experiments.
      ).
      For non-linearly oscillating bubbles, attenuation can be modeled by solving the full non-linear RP equation to obtain the radius time curves of all bubbles in the size distribution. Then the scattered pressure can be calculated, using eqn (9), and subsequently the scattering cross section σs. Attenuation is then calculated though integration over the size distribution (
      • Hilgenfeldt S.
      • Lohse D.
      • Zomack M.
      Sound scattering and localized heat deposition of pulse-driven microbubbles.
      ;
      • Segers T.
      • de Jong N.
      • Versluis M.
      Uniform scattering and attenuation of acoustically sorted ultrasound contrast agents: Modeling and experiments.
      ). Note that for contrast bubbles, Ps is dependent not only on the ultrasound frequency but also on the acoustic driving pressure. As a consequence, the acoustic bubble response will depend not only on the local pressure at the bubble position in the transmit beam (typically extracted from hydrophone measurements in a calibration tank) but also on the attenuation experienced by the transmit beam on its path to the bubble, which will lower the local acoustic driving pressure. Thus, to accurately model attenuation of a sample of contrast bubbles, the full two-way coupled interaction of the non-linearly oscillating bubbles with the complete ultrasound field must be taken into account (e.g., through forward integration) (
      • Segers T.
      • Kruizinga P.
      • Kok M.
      • Lajoinie G.
      • de Jong N.
      • Versluis M.
      Monodisperse versus polydisperse ultrasound contrast agents: Non-linear response, sensitivity, and deep tissue imaging potential.
      ). For acoustic characterization of UCAs, the effect of a non-uniform acoustic driving pressure can be minimized by using a narrow bubble sample holder such that it forms a sheet of bubbles in the acoustic characterization beam rather than a 3-D volume (
      • Segers T.
      • Kruizinga P.
      • Kok M.
      • Lajoinie G.
      • de Jong N.
      • Versluis M.
      Monodisperse versus polydisperse ultrasound contrast agents: Non-linear response, sensitivity, and deep tissue imaging potential.
      ).

      Bubble non-sphericity and asymmetry

      While spherical-bubble models provide a basic fundamental understanding of bubble behavior and the influence of the physical parameters of the surrounding environment, the assumption of spherical symmetry is often challenged. Then, modeling the bubble behavior requires the inclusion of the effects of surface instabilities and neighboring membranes or walls. This breaking of symmetry deeply affects the fundamental behavior of a bubble, namely its resonance frequency and quality factor, and enables, or amplifies, higher-order phenomena such as streaming.
      Two main mechanisms can destabilize the spherical shape of a bubble during ultrasound-induced oscillations in the free field: Rayleigh–Taylor–type instability and parametric instability. Rayleigh–Taylor instability can occur at the point of maximum bubble compression while the bubble wall is experiencing maximum acceleration. It can only dominate after repeated, strong collapse events of the bubble or during the after-bounces that follow violent bubble collapse occurring at a low driving frequency (e.g., in sonoluminescence) (
      • Hilgenfeldt S.
      • Lohse D.
      • Brenner M.P.
      Phase diagrams for sonoluminescing bubbles.
      ). In such a configuration, shape destabilization results in pinch-off, jetting or bubble breakup, and the bubble fate becomes unpredictable. Parametric instabilities, on the other hand, arise during long driving pulses. They grow from an infinitesimally small perturbation on the bubble interface and develop with certain mode preferences depending on the bubble size and driving frequency. Such instabilities are often observed for UCAs (see a typical example in Fig. 8a;
      • Raymond J.L.
      • Luan Y.
      • Peng T.
      • Huang S.L.
      • McPherson D.D.
      • Versluis M.
      • de Jong N.
      • Holland C.K.
      Loss of gas from echogenic liposomes exposed to pulsed ultrasound.
      ), especially when using long driving pulses (e.g., for therapeutic applications).
      Fig 8
      Fig. 8(a) Evolution of the buildup of surface mode vibrations for echogenic liposomes. Asymmetric patterns indicative of surface mode oscillations are first visible, after several cycles of acoustic excitation (copyright Institute of Physics and Engineering in Medicine. Reproduced by permission of IOP Publishing from
      • Raymond J.L.
      • Luan Y.
      • Peng T.
      • Huang S.L.
      • McPherson D.D.
      • Versluis M.
      • de Jong N.
      • Holland C.K.
      Loss of gas from echogenic liposomes exposed to pulsed ultrasound.
      . All rights reserved). (b) Surface mode vibrations for BR-14 (Bracco Research) contrast agent. (Top) Applied acoustic pressure; (middle) radius-time curve; (bottom) evolution of mode 2, which saturates after about 4 cycles. Note the period-doubling behavior of the surface mode vibration (reproduced with permission from
      • Dollet B.
      • van der Meer S.M.
      • Garbin V.
      • de Jong N.
      • Lohse D.
      • Versluis M.
      Nonspherical oscillations of ultrasound contrast agent microbubbles.
      ). (c) Top-view and side-view ultrahigh-speed imaging of ultrasound-driven contrast microbubbles (reproduced with permission from
      • Vos H.J.
      • Dollet B.
      • Versluis M.
      • de Jong N.
      Nonspherical shape oscillations of coated microbubbles in contact with a wall.
      ). (d) Shift in the resonance frequency of a cylindrical bubble of length l as a function of initial length l0 and axial location e within an open-ended tube of length L and radius Rtube (adapted from
      • Oguz H.N.
      • Prosperetti A.
      The natural frequency of oscillation of gas bubbles in tubes.
      ). In the top panel, L/Rtube = 10 and e/L = 0.5; in the bottom panel, L/Rtube = 10 and l/Rtube = 0.17.

      Non-spherical oscillations: Modeling

      One of the first models for spherical instability was developed by
      • Plesset M.S.
      On the stability of fluid flows with spherical symmetry.
      . In essence, the derivation considers a nearly spherical bubble, with a distortion of small amplitude from the spherical shape that can be written in the form of spherical harmonics:
      R(t,θ,ϕ)=R(t)+an(t)Yn(θ,ϕ),
      (18)


      with Yn the spherical harmonic mode of order n and an its amplitude; R(t) is assumed to obey the spherical RP equation (eqn [1]). Using an analogous derivation of the RP equation, and following a first-order Taylor expansion, we find
      a¨n+3R˙Ra˙n((n1)R¨R(n1)(n+1)(n+2)σρLR3)an=0.
      (19)


      As before, the overdots represent the time derivatives. This equation is a Hill equation, or a generalized Mathieu equation. The eigenfrequency of the mode directly follows from the small oscillation limit:
      ω0n=(n1)(n+1)(n+2)σρLR03.
      (20)


      In this potential flow approach, the velocity field is described by the gradient of the velocity potential ∇ϕ, which is a function of space and time. This description, however, does not include any form of damping or viscosity, which results in parametric mode behavior with a threshold of R˙=0. This model was later picked up by
      • Ceschia M.
      • Nabergoj R.
      On the motion of a nearly spherical bubble in a viscous liquid.
      and
      • Nabergoj R.
      • Francescutto A.
      On thresholds for surface waves on resonant bubbles.
      , who applied a similar principle by using a Lagrangian approach to find the equation of motion of the bubble from the RP equation and the governing equation of the modes. They assumed purely sinusoidal bubble oscillations; the vorticity is neglected and the viscosity is introduced solely by assuming a vanishing tangential stress. Note that this last assumption is debatable for coated microbubbles, owing to the presence of the shell. This derivation, using a known solution for the resulting Mathieu equation, provides a first estimate of the threshold for the growth of each parametric mode (i.e., the pressure leading to bubble instability). Despite these simplifications, the results have demonstrated good agreement with experimental observations (Fig. 8b;
      • Dollet B.
      • van der Meer S.M.
      • Garbin V.
      • de Jong N.
      • Lohse D.
      • Versluis M.
      Nonspherical oscillations of ultrasound contrast agent microbubbles.
      ;
      • Versluis M.
      • Goertz D.E.
      • Palanchon P.
      • Heitman I.L.
      • van der Meer S.M.
      • Dollet B.
      • de Jong N.
      • Lohse D.
      Microbubble shape oscillations excited through ultrasonic parametric driving.
      ).
      Around the same time,
      • Prosperetti A.
      Thermal effects and damping mechanisms in the forced radial oscillations of gas bubbles in liquids.
      provided a more rigorous theoretical framework to study surface oscillations of a bubble. Vorticity was accounted for by writing the curl as the sum of a poloidal and a toroidal component. Application to a non-driven bubble provides the same mode frequency and decay rate as given by
      • Nabergoj R.
      • Francescutto A.
      On thresholds for surface waves on resonant bubbles.
      . Based on the reference work of Prosperetti, subsequent studies (
      • Hilgenfeldt S.
      • Lohse D.
      • Brenner M.P.
      Phase diagrams for sonoluminescing bubbles.
      ;
      • Wu C.C.
      • Roberts P.H.
      Bubble shape instability and sonoluminescence.
      ) have lifted some assumptions on the vorticity and instead limited the range of the curl to a small boundary layer whose thickness δBL depends on the bubble size and the mode number:
      δBL=min(μLρLω,R02n).
      (21)


      Including this short-range vorticity modifies the second term of eqn (19), where 3R˙/R must be replaced by the damping term Bn(t) given by eqn (13) of
      • Hilgenfeldt S.
      • Lohse D.
      • Brenner M.P.
      Phase diagrams for sonoluminescing bubbles.
      .
      Two major research directions evolved from this founding work. First, the coupling between bubble motion and surface modes is of particular importance for local drug delivery that involves bubble translation and bubble oscillations. This problem has been mainly investigated in an axisymmetric geometry with viscous effects (
      • Benjamin T.B.
      • Ellis A.T.
      Self-propulsion of asymmetrically vibrating bubbles.
      ;
      • Shaw S.J.
      The stability of a bubble in a weakly viscous liquid subject to an acoustic traveling wave.
      ) or without (
      • Shaw S.J.
      Translation and oscillation of a bubble under axisymmetric deformation.
      ), demonstrating that shape oscillations can result in self-propulsion and bubble translation. This phenomenon is distinct from radiation forces. Some attention has also been given to the coupling between the surface modes themselves (
      • Mei C.C.
      • Zhou X.
      Parametric resonance of a spherical bubble.
      ;
      • Feng Z.C.
      • Leal L.G.
      On energy transfer in resonant bubble oscillations.
      ;
      • Mao Y.
      • Crum L.A.
      • Roy R.A.
      Nonlinear coupling between the surface and volume modes of an oscillating bubble.
      ;
      • Harkin A.A.
      • Kaper T.J.
      • Nadim A.
      Energy transfer between the shape and volume modes of a nonspherical gas bubble.
      ).
      The second direction consists in modeling the impact of the shell on the existence of surface modes. A few recent studies have investigated the effect of an elastic shell on the parametric oscillations of a coated bubble (
      • Liu Y.
      • Sugiyama K.
      • Takagi S.
      • Matsumoto Y.
      Surface instability of an encapsulated bubble induced by an ultrasonic pressure wave.
      ;
      • Liu Y.
      • Wang Q.
      Stability and natural frequency of nonspherical mode of an encapsulated microbubble in a viscous liquid.
      ;
      • Liu Y.
      • Calvisi M.L.
      • Wang Q.
      Shape oscillation and stability of an encapsulated microbubble translating in an acoustic wave.
      ). These articles consider linear elasticity theory for the shell, with or without the effect of viscosity, and have determined a correction for the eigenfrequency of the surface modes to account for the presence of the elastic shell. However, polymer-coated ultrasound contrast microbubbles typically fail to display such extreme elastic behavior owing to the buckling or breakup of the fairly rigid shell (Fig. 1h;
      • Marmottant P.
      • Bouakaz A.
      • de Jong N.
      • Quilliet C.
      Buckling resistance of solid shell bubbles under ultrasound.
      ). The influence of a non-linear phospholipid shell, which constitutes the vast majority of contrast microbubble shells, remains mostly unexplored.

      Bubbles near a wall

      The presence of a boundary near the oscillating bubbles is difficult to avoid, both in vitro and in vivo, and is known to influence the response of the microbubble in a non-trivial way, affecting its oscillation amplitude and non-linear response and shifting its resonance frequency. When the bubble is targeted, and therefore adherent to the wall, bubble–wall interactions are inevitable and typically desirable. It is therefore of utmost importance to fundamentally understand the effect of a neighboring wall on the bubble response. In vivo, it is key to enabling many novel applications that require a very fine control of bubble dynamics. In vitro, it is essential to extract meaningful and precise mechanical parameters describing the bubble and its shell as well as its interaction with the boundary.
      Physically, the presence of a wall has two distinct effects whose magnitudes depend on its mechanical properties: (i) the bubble interacts with its own acoustic reflection returning from the interface, often described as the image bubble, and (ii) the flow around the bubble deviates from the spherical flow derived from the RP equation, owing to the presence of the wall.
      The simplest treatment of the boundary was proposed by
      • Lamb H.
      Hydrodynamics.
      and consists in adding a pressure term to the RP equation representing the pressure reflected at the interface. Since the bubble is considered to be small compared with the acoustic wavelength, phase lag is disregarded. The reflection decreases the resonance frequency by a factor of 2/3. Much later,
      • Doinikov A.A.
      • Aired L.
      • Bouakaz A.
      Acoustic scattering from a contrast agent microbubble near an elastic wall of finite thickness.
      investigated the problem from a potential flow perspective. They considered a thin flexible wall and added bubble translation as a result of radiation forces. In this picture, the resonance frequency decreases for a rigid wall and increases for an elastic wall, compared with the free-bubble case. This model was later confirmed experimentally by the same group (
      • Doinikov A.A.
      • Bouakaz A.
      Interaction of an ultrasound-activated contrast microbubble with a wall at arbitrary separation distances.
      ). This result is well aligned with that of
      • Oguz H.N.
      • Prosperetti A.
      Bubble oscillations in the vicinity of a nearly plane free surface.
      , who used perturbation theory to derive a model describing the extreme case of a bubble oscillating near a free surface. That work shows that, near a free surface, the resonance frequency should increase by about 40%.
      With respect to microbubble oscillations near a wall, of special interest are the non-linear echoes of the microbubbles. These non-linearities are essential for harmonic imaging, including pulse inversion and amplitude modulation, designed to boost the contrast-to-tissue ratio.
      • Suslov S.A.
      • Ooi A.
      • Manasseh R.
      Nonlinear dynamic behavior of microscopic bubbles near a rigid wall.
      have shown that the dominant harmonic frequencies that are emitted depend on the bubble-to-wall distance. They also suggest an increased chance of period-doubling behavior for bubbles in contact with the wall. The most important effect of a boundary on the bubble behavior, however, lies in the loss of spherical symmetry of the bubble oscillations. Such non-spherical oscillations are systematically observed experimentally (e.g., Fig. 8c), but they are generally overlooked in the modeling work. Recently,
      • Lajoinie G.
      • Luan Y.
      • Gelderblom E.
      • Dollet B.
      • Mastik F.
      • Dewitte H.
      • Lentacker I.
      • de Jong N.
      • Versluis M.
      Non-spherical oscillations drive the ultrasound-mediated release from targeted microbubbles.
      proposed a model for bubbles adherent to or in contact with a boundary, based on a segmentation of the bubble along the inclination angle and with the bubble dynamics solved on a per-segment basis. Despite its extreme simplicity, this model recovers the non-spherical shapes observed experimentally and quantifies the effect of the boundary on the bubble oscillation amplitude and subsequently on streaming (discussed later under “Modeling of higher-order effects”). Much work, however, remains to be done to achieve a rigorous description of non-spherical bubble oscillations near a boundary, especially in relation to resonance shift, and of non-linear emissions.

      Bubbles in a tube

      As blood-pool agents, contrast microbubbles circulate within the confines of the blood vessel network (i.e., within tubes of various sizes lined with compliant walls). The simplified 1-D case of a bubble of length l oscillating without surface tension or friction in the center of a rigid tube of length L yields (
      • Sun C.
      • Can E.
      • Dijkink R.
      • Lohse D.
      • Prosperetti A.
      Growth and collapse of a vapour bubble in a microtube: The role of thermal effects.
      ):
      ρLl¨(Ll)=P0(l0l)κP(L).
      (22)


      After linearization around l=l0(1+ε) for small amplitude  ε  ≪ 1,
      [ρLl0(Ll0)]ε¨+κP0ε=PA,
      (23)


      with eigenfrequency
      ω0=κP0ρLl0(Ll0).
      (24)


      Owing to the incompressibility of the liquid, it is clear that significant pulsations are only possible if the tube has an open end.
      • Oguz H.N.
      • Prosperetti A.
      The natural frequency of oscillation of gas bubbles in tubes.
      proposed a more complete incompressible disk bubble model that includes the effects of surface tension, the position e of the bubble along the tube axis, and tube radius Rtube (Fig. 8d). This model also provides a first-order estimate for the thermal and viscous damping experienced by such a bubble.
      In the same paper,
      • Oguz H.N.
      • Prosperetti A.
      The natural frequency of oscillation of gas bubbles in tubes.
      also proposed a model for small bubbles obtained by projecting the velocity potential in the liquid on to the Legendre polynomial base. According to this model, a bubble 10 times smaller than the tube is still strongly affected by the geometry, even when the tube in question is short. Going further,
      • Leighton T.G.
      The inertial terms in equations of motion for bubbles in tubular vessels or between plates.
      has shown, by changing the inertial term in the RP equation, that a long rigid tube 1000 times larger than the bubble can have a significant effect on its behavior.
      Let us now consider a bubble trapped in a long tube, replicating a vessel, with a diameter of the same order as the bubble size. If we assume that the compressibility of the surrounding medium does not allow for large-amplitude oscillations (e.g., in water), significant bubble motion can only arise from large compliance of the vessel wall, which allows for bubble motion in three dimensions. The compliance of a blood vessel and its response to stress are complex to model in themselves (
      • Misra J.C.
      • Singh S.I.
      A large deformation analysis for aortic walls under a physiological loading.
      ;
      • Humphrey J.D.
      • Na S.
      Elastodynamics and arterial wall stress.
      ), in particular since blood vessels behave in a non-linear fashion.
      • Qin S.
      • Ferrara K.W.
      Acoustic response of compliable microvessels containing ultrasound contrast agents.
      have proposed a model that describes the axisymmetric bubble oscillations in a non-linear compliant vessel by adding a pressure jump across the vessel-wall location. There, the properties of the blood vessel are fitted to stress measurements on frog mesenteric capillaries. An important conclusion of this model is that, on the one hand, the bubble behavior is modified by the presence of the wall, and on the other hand, the stress experienced by the vessel wall is far greater if a bubble is present within the system during insonation. This model was later amended by
      • Qin S.
      • Ferrara K.W.
      The natural frequency of nonlinear oscillation of ultrasound contrast agents in microvessels.
      and numerically solved to include the effect of soft tissue around the vessel, providing an extensive study of the impact of wall compliance and its geometric parameters. This line of modeling is of major importance for applications such as the controlled opening of the blood–brain barrier. Owing to the complexity of such modeling, most investigations are performed numerically, but new biomedical applications of bubbles that require more precision and control would also benefit from simplified analytical models.

      Bubble jetting

      When a bubble undergoes collapse in an asymmetric environment, it forms a jet. The asymmetry can result from the presence of a nearby soft (or gaseous) interface (
      • Krishnan S.
      • Hopfinger E.J.
      • Puthenveettil B.A.
      On the scaling of jetting from bubble collapse at a liquid surface.
      ), a rigid boundary, other bubbles in the vicinity (
      • Han B.
      • Köhler K.
      • Jungnickel K.
      • Mettin R.
      • Lauterborn W.
      • Vogel A.
      Dynamics of laser-induced bubble pairs.
      ;
      • Kwan J.J.
      • Lajoinie G.
      • de Jong N.
      • Stride E.
      • Versluis M.
      • Coussios C.C.
      Ultrahigh-speed dynamics of micrometer-scale inertial cavitation from nanoparticles.
      ) or even a non-linear acoustic wave (
      • Ohl C.D.
      • Ikink R.
      Shock-wave-induced jetting of micron-size bubbles.
      ). Jet formation can have severe consequences; the high jet velocity is crucial in bubble-induced erosion and in bubble cleaning applications (
      • Fernandez Rivas D.
      • Verhaagen B.
      • Seddon J.R.T.
      • Zijlstra A.G.
      • Jiang L.M.
      • van der Sluis L.W.M.
      • Versluis M.
      • Lohse D.
      • Gardeniers H.J.G.E.
      Localized removal of layers of metal, polymer, or biomaterial by ultrasound cavitation bubbles.
      ;
      • Verhaagen B.
      • Fernandez Rivas D.
      Measuring cavitation and its cleaning effect.
      ). Jetting can lead to the breakup of bubbles into numerous daughter bubbles, the destruction of the agent, or the deposition of the shell material onto the rigid boundary (
      • Roovers S.
      • Lajoinie G.
      • Cock I.D.
      • Brans T.
      • Dewitte H.
      • Braeckmans K.
      • Versluis M.
      • Smedt S.C.D.
      • Lentacker I.
      Sonoprinting of nanoparticle-loaded microbubbles: Unraveling the multi-timescale mechanism.
      ). Jetting becomes more prominent as bubble size increases (owing to reduced stabilization by surface tension), as frequency decreases and as pressure increases (owing to increased inertia).
      In terms of fluid dynamics, bubble jetting can be described using fairly strong assumptions: the flow is considered inviscid and irrotational, leading to a potential flow description. This approach was first presented by
      • Plesset M.S.
      • Chapman R.B.
      Collapse of an initially spherical vapour cavity in the neighbourhood of a solid boundary.
      , who predicted jet velocities on the order of 100 m/s. The shapes that were computed based on this approach were later experimentally verified by
      • Lauterborn W.
      • Bolle H.
      Experimental investigations of cavitation-bubble collapse in the neighbourhood of a solid boundary.
      (Fig. 9a) with remarkable agreement. Going beyond this initial description,
      • Blake J.R.
      • Tomita Y.
      • Tong R.P.
      The art, craft and science of modelling jet impact in a collapsing cavitation bubble.
      implemented the presence of the boundary using a Green's function for the velocity potential to account for the source (the bubble) and its image. This Green's function enforces a no-flow condition across the boundary.
      Fig 9
      Fig. 9Bubble jetting. (a) Calculated shape of a jetting bubble (reproduced with permission from
      • Lauterborn W.
      • Bolle H.
      Experimental investigations of cavitation-bubble collapse in the neighbourhood of a solid boundary.
      ); (b) jetting observed for a microbubble and a schematic representation of jet formation (reproduced with permission from
      • Postema M.
      • van Wamel A.
      • ten Cate F.J.
      • de Jong N.
      High-speed photography during ultrasound illustrates potential therapeutic applications of microbubbles.
      ); (c) jetting observed for a 3.0 μm lipid-coated bubble (reproduced with permission from
      • Lajoinie G.
      • Luan Y.
      • Gelderblom E.
      • Dollet B.
      • Mastik F.
      • Dewitte H.
      • Lentacker I.
      • de Jong N.
      • Versluis M.
      Non-spherical oscillations drive the ultrasound-mediated release from targeted microbubbles.
      ); (d) SonoVue ultrasound contrast microbubbles driven at 200 kHz jetting in a capillary, with ultrasound beam (red dashed line; reproduced with permission from
      • Cleve S.
      • Inserra C.
      • Prentice P.
      Contrast agent microbubble jetting during initial interaction with 200-kHz focused ultrasound.
      ).
      To better account for the large jet velocities involved and the presence of an acoustic wave,
      • Wang Q.X.
      • Blake J.R.
      Non-spherical bubble dynamics in a compressible liquid. Part 1: travelling acoustic wave.
      extended these formulations to slightly compressible fluids, demonstrating a measurable effect of compressibility on bubble jetting and in particular on the orientation of the jet as compared with the propagation direction of the ultrasound wave. Next to bubble size, ultrasound frequency and pressure, jetting is also very sensitive to the initial distance between the bubble and the rigid boundary (Fig. 9b–d)(
      • Prentice P.
      • Cuschieri A.
      • Dholakia K.
      • Prausnitz M.
      • Campbell P.
      Membrane disruption by optically controlled microbubble cavitation.
      ;
      • Wang Q.
      • Liu W.
      • Zhang A.M.
      • Sui Y.
      Bubble dynamics in a compressible liquid in contact with a rigid boundary.
      ). In particular, the problem was treated by
      • Brujan E.-A.
      Jets from pulsed-ultrasound-induced cavitation bubbles near a rigid boundary.
      using weakly compressible theory, showing that jet velocity rapidly increases as the initial bubble–boundary distance increases, while the total amount of kinetic energy in the jet overall decreases.
      Biological tissues present a wide range of mechanical properties but are nearly all flexible in the sense that they can be deformed during bubble oscillations. This is true for cell membranes as well as for blood-vessel walls. It is therefore of prime importance to describe the effect of soft surfaces on jetting, which is expected to be a major cause of damage in vivo. It has been shown that the jet direction can be reversed when the substrate material becomes very soft, namely when it becomes easier to pull the elastic material than to overcome the water inertia (
      • Gibson D.C.
      • Blake J.R.
      The growth and collapse of bubbles near deformable surfaces.
      ;
      • Brujan E.A.
      • Nahen K.
      • Schmidt P.
      • Vogel A.
      Dynamics of laser-induced cavitation bubbles near elastic boundaries: Influence of the elastic modulus.
      ;
      • Li S.
      • Zhang A.M.
      • Han R.
      Letter: Counter-jet formation of an expanding bubble near a curved elastic boundary.
      ). There have, however, been only a few studies on the transition regimes and on its effect in biological tissues (
      • Chen H.
      • Kreider W.
      • Brayman A.A.
      • Bailey M.R.
      • Matula T.J.
      Blood vessel deformations on microsecond time scales by ultrasonic cavitation.
      ).

      Modeling bubbles in tissue and soft matter

      Given the complex nature of biological tissues and the interest in novel extravasating agents and bubble precursors, it is important to study the behavior of bubbles in viscoelastic materials. In particular, the bubble resonance frequency is known to change significantly with the elastic modulus of the enclosing medium. A number of mechanical models have been developed—initially based on Kelvin–Voigt models (
      • Macosko C.W.
      Rheology: Principles, measurements, and applications.
      ) and later on Maxwell models, both linear (
      • Allen J.S.
      • Roy R.A.
      Dynamics of gas bubbles in viscoelastic fluids. I. Linear viscoelasticity.
      ) and non-linear (
      • Allen J.S.
      • Roy R.A.
      Dynamics of gas bubbles in viscoelastic fluids. II. Nonlinear viscoelasticity.
      ;
      • Naude J.
      • Mendez F.
      Periodic and chaotic acoustic oscillations of a bubble gas immersed in an upper convective Maxwell fluid.
      )—demonstrating the change in resonance frequency as a result of the elastic modulus of the bubble surroundings.
      • De Corato M.
      • Saint-Michel B.
      • Makrigiorgos G.
      • Dimakopoulos Y.
      • Tsamopoulos J.
      • Garbin V.
      Oscillations of small bubbles and medium yielding in elastoviscoplastic fluids.
      investigated the dynamics of large bubbles (>50 μm) immersed in a yield-stress fluid using theory and numerical simulations. They quantified the viscoelastic and viscoplastic response of the material using the model developed by
      • Saramito P.
      A new elastoviscoplastic model based on the Herschel–Bulkley viscoplastic model.
      . In another notable example,
      • Yang X.
      • Church C.C.
      A model for the dynamics of gas bubbles in soft tissue.
      investigated the increased threshold for subharmonic emissions in these materials. Linear and non-linear viscoelastic models are more extensively reviewed by
      • Dollet B.
      • Marmottant P.
      • Garbin V.
      Bubble dynamics in soft and biological matter.
      .

      Modeling higher-order effects

      Streaming and shedding

      Acoustic streaming is a phenomenon whereby an oscillating flow generates a non-linear, second-order steady flow (Fig. 10a, 10b). In particular, it occurs in the presence of walls, along which viscous effects are localized in a boundary layer of typical thickness δBL=μL/ρLω, with ω the angular frequency.
      • Rayleigh L.
      On the circulation of air observed in Kundt's tubes, and on some allied acoustical problems.
      was the first to describe acoustic streaming theoretically. He showed that an oscillatory flow of velocity U(x)cos ωt along a wall (x being a coordinate along the wall) causes a steady-state streaming flow of velocity us such that us(x)=(3/4)U(x)U(x)/ω along the wall (e.g.,
      • Lighthill J.
      Acoustic streaming.
      ); to be rigorous, this applies only if the streaming Reynolds number usρL/μL is much lower than 1 ( ℓ being the typical scale over which streaming takes place).
      Fig 10
      Fig. 10(a) Microstreaming field as a result of an oscillating SonoVue microbubble (reproduced with permission from
      • Pereno V.
      • Aron M.
      • Vince O.
      • Mannaris C.
      • Seth A.
      • de Saint Victor M.
      • Lajoinie G.
      • Versluis M.
      • Coussios C.
      • Carugo D.
      • Stride E.
      Layered acoustofluidic resonators for the simultaneous optical and acoustic characterisation of cavitation dynamics, microstreaming, and biological effects.
      ). (b) Lipid shedding and transport as a result of acoustic streaming (reproduced with permission from
      • Versluis M.
      High-speed imaging in fluids.
      ). (c) Simulation of the acoustic streaming generated by an oscillating bubble of radius a (the half-circle) in contact with a plane wall located at z = 0. Only half of the space is sketched, owing to axisymmetry about the x = 0 axis. The solid curves are the streamlines, and the direction of velocity is indicated with the arrows; in particular, the streaming flow goes away from the wall above the bubble (reproduced with permission from
      • Marmottant P.
      • Versluis M.
      • de Jong N.
      • Hilgenfeldt S.
      • Lohse D.
      High-speed imaging of an ultrasound-driven bubble in contact with a wall: “Narcissus” effect and resolved acoustic streaming.
      ). (d) Lipid shedding from a targeted bubble and transport by acoustic streaming. (Top) Schematics of the shedding process of the released material (red) with the calculated streamlines (solid black lines); (bottom) experimental recording of the release of the fluorescent material from a targeted oscillating microbubble driven at a pressure of 331 kPa, showing transport over a distance several times larger than the bubble size (reproduced with permission from
      • Lajoinie G.
      • Luan Y.
      • Gelderblom E.
      • Dollet B.
      • Mastik F.
      • Dewitte H.
      • Lentacker I.
      • de Jong N.
      • Versluis M.
      Non-spherical oscillations drive the ultrasound-mediated release from targeted microbubbles.
      ). (E) Streak imaging of two ultrasound contrast agents of equilibrium radius 2.1 μm (top) and 2.0 μm (bottom), showing the displacement in the direction of propagation of the acoustic wave. The acoustic forcing represents 20 cycles at amplitude 380 kPa and frequency 2.25 MHz. These snapshots are overlaid with simulated curves of the time evolution of the bubble radius (reproduced with permission from
      • Dayton P.A.
      • Allen J.S.
      • Ferrara K.W.
      The magnitude of radiation force on ultrasound contrast agents.
      ). (F) Evolution of the radius of a lipid-coated microbubble subjected to repeated acoustic pulses (frequency 2.25 MHz, peak negative pressure 400 kPa, pulse repetition frequency 0.5 Hz; data kindly provided by Mark Borden).
      Acoustic streaming can be generated by an oscillating bubble close to a wall (
      • Marmottant P.
      • Hilgenfeldt S.
      Controlled vesicle deformation and lysis by single oscillating bubbles.
      ) (Fig. 10c). It is thought to be a key transport mechanism for drug release from drug-loaded UCAs (
      • Luan Y.
      • Lajoinie G.
      • Gelderblom E.
      • Skachkov I.
      • van der Steen A.F.W.
      • Vos H.J.
      • Versluis M.
      • de Jong N.
      Lipid shedding from single oscillating microbubbles.
      ), and one of the main mechanisms of sonoporation through the induced shear stress on a cell layer. To theoretically assess the latter effect in the specific context of UCAs,
      • Doinikov A.A.
      • Bouakaz A.
      Theoretical investigation of shear stress generated by a contrast microbubble on the cell membrane as a mechanism for sonoporation.
      computed the shear stress associated with streaming. Generally speaking, it is of order τxz ≈ μLus/δBL, z being the coordinate away from the wall. Hence, τxzρLμL/ωU(x)U(x).
      • Doinikov A.A.
      • Bouakaz A.
      Theoretical investigation of shear stress generated by a contrast microbubble on the cell membrane as a mechanism for sonoporation.
      also computed the velocity U related to a microbubble located at a distance d from the wall and undergoing volumetric oscillations of radius R(t) forced by an acoustic field, using a modified RP equation with the model by
      • de Jong N.
      • Cornet R.
      • Lancée C.T.
      Higher harmonics of vibrating gas-filled microspheres, part one: Simulations.
      for the shell term, and obtained the corresponding shear stress. They showed that it displays a maximum at a finite radial distance along the wall, and estimated its potential implications for sonoporation.
      • Chomas J.E.
      • Dayton P.
      • Allen J.
      • Morgan K.
      • Ferrara K.W.
      Mechanisms of contrast agent destruction.
      presented one of the first systematic studies of UCA destruction under insonation. The most violent destruction mechanism is fragmentation into smaller bubbles at high amplitude, but at lower amplitude they evidenced a progressive deflation of the UCA, which they argue is owing to acoustically driven diffusion. Actually, when lipid monolayers get sufficiently compressed, they can shed some material in the surrounding liquid. This mechanism has been recognized as important in the context of the slow dissolution of lipid-coated bubbles (
      • Borden M.A.
      • Longo M.L.
      Dissolution behavior of lipid monolayer-coated, air-filled microbubbles: Effect of lipid hydrophobic chain length.
      ) (Fig. 1i) and has been proposed as key for the understanding of UCA deflation under prolonged insonation. Using fluorescence imaging,
      • Borden M.A.
      • Kruse D.E.
      • Caskey C.F.
      • Zhao S.
      • Dayton P.A.
      • Ferrara K.W.
      Influence of lipid shell physicochemical properties on ultrasound-induced microbubble destruction.
      evidenced ultrasound-induced shedding and quantified the time evolution of bubble deflation toward an equilibrium (around 2 μm in radius) (Fig. 10f). They showed different mechanisms of shedding, depending on lipid composition, from submicrometer particle expulsion to budding, then to buildup and expulsion of larger lipid aggregates as the shell becomes more cohesive. Similar observations have been made by
      • Kooiman K.
      • van Rooij T.
      • Qin B.
      • Mastik F.
      • Vos H.J.
      • Versluis M.
      • Klibanov A.L.
      • de Jong N.
      • Villanueva F.S.
      • Chen X.
      Focal areas of increased lipid concentration on the coating of microbubbles during short tone-burst ultrasound insonification.
      using ultrahigh-speed imaging.
      • Kwan J.J.
      • Borden M.M.
      Lipid monolayer collapse and microbubble stability.
      argue that the microscopic mechanisms initiating shedding are the local collapse of the lipid monolayer and the subsequent aggregation of new folds.
      • Guidi F.
      • Vos H.J.
      • Mori R.
      • de Jong N.
      • Tortoli P.
      Microbubble characterization through acoustically induced deflation.
      subjected UCAs to repeated low-amplitude ultrasonic pulses at a frequency of 2 MHz, and consistently observed deflation for bubbles with an initial radius below 3 μm toward a stable radius of 1.4 μm, but no deflation for larger bubbles.
      The time evolution of the deflation process is qualitatively similar to the one reported by
      • Borden M.A.
      • Kruse D.E.
      • Caskey C.F.
      • Zhao S.
      • Dayton P.A.
      • Ferrara K.W.
      Influence of lipid shell physicochemical properties on ultrasound-induced microbubble destruction.
      , with the fastest deflation rate near resonance, where UCAs are more prone to dissolve or shed their coating owing to large volumetric oscillations.
      • Viti J.
      • Mori R.
      • Guidi F.
      • Versluis M.
      • de Jong N.
      • Tortoli P.
      Nonlinear oscillations of deflating bubbles.
      focused on megahertz bubble oscillations using ultrahigh-speed imaging while the bubbles underwent (slower) deflation. They showed that deflation occurs in discrete steps, and that the radius time oscillations usually evolve from being rather symmetric just after a deflation event to compression-only behavior after some time in a non-deflating state, thereby suggesting that the lipid shell reforms after each shedding event.
      The idea that a sufficiently compressed lipid monolayer collapses and expels excess lipids is the basis of the surfactant-shedding model of
      • O'Brien J.
      • Ovenden N.
      • Stride E.
      Accounting for the stability of microbubbles to multi-pulse excitation using a lipid-shedding model.
      , based on phenomenological dependences of molecular diffusivity and surface viscosity on surface concentration of lipids. This model reproduced the deflation results of
      • Borden M.A.
      • Kruse D.E.
      • Caskey C.F.
      • Zhao S.
      • Dayton P.A.
      • Ferrara K.W.
      Influence of lipid shell physicochemical properties on ultrasound-induced microbubble destruction.
      . To clarify lipid shedding dynamics under insonation,
      • Luan Y.
      • Lajoinie G.
      • Gelderblom E.
      • Skachkov I.
      • van der Steen A.F.W.
      • Vos H.J.
      • Versluis M.
      • de Jong N.
      Lipid shedding from single oscillating microbubbles.
      used high-speed fluorescence imaging at 150,000 frames/s. They showed that shedding may occur already within a few ultrasound pulses, in contrast to the picture of a (relatively slow) build up of expelled material.
      • Lajoinie G.
      • Luan Y.
      • Gelderblom E.
      • Dollet B.
      • Mastik F.
      • Dewitte H.
      • Lentacker I.
      • de Jong N.
      • Versluis M.
      Non-spherical oscillations drive the ultrasound-mediated release from targeted microbubbles.
      refined this setup by coupling it with ultrahigh-speed imaging. They showed that non-spherical oscillations of a lipid-coated bubble in contact with a wall lead to local oversaturation and shedding, and that the subsequent transport of the shed lipid is mediated by acoustic streaming (Fig. 10d).

      Radiation forces

      Any body of volume V in a space-dependent pressure field p experiences a force equal to Vp, provided that the size of the body is much smaller than the acoustic wavelength. If the body is a bubble in an acoustic field, this can lead to a strong time-averaged force, owing to the large gas compressibility. This force is termed acoustic radiation force or, in the bubble context, Bjerknes force (
      • Bjerknes V.F.K.
      Fields of force: Supplementary lectures, applications to meteorology.
      ;
      • Crum L.A.
      Bjerknes forces on bubbles in a stationary sound field.
      ;
      • Leighton T.G.
      The acoustic bubble.
      ). Two radiation forces are usually distinguished: primary forces, when the bubble responds directly to the external applied pressure field (typically from a transducer), and secondary forces, when a given bubble reacts to the pressure field re-emitted by a second neighboring bubble or, in the case of a neighboring wall, by the image bubble. These forces are maximal at the resonance frequency of the bubble. For traveling waves, the primary radiation force always acts in the direction of ultrasound wave propagation.
      • Dayton P.A.
      • Allen J.S.
      • Ferrara K.W.
      The magnitude of radiation force on ultrasound contrast agents.
      characterized the influence of primary radiation forces on UCAs. They showed that a bubble with a radius of 1.5 μm can be displaced by as much as 5 μm by a 20-cycle pulse with a frequency of 2.25 MHz at an amplitude of 180 kPa (Fig. 10E). However, assessing radiation forces quantitatively turns out to be difficult because the force balance involves hydrodynamic forces, including added mass, drag and history force.
      • Dayton P.
      • Klibanov A.
      • Brandenburger G.
      • Ferrara K.
      Acoustic radiation force in vivo: A mechanism to assist targeting of microbubbles.
      ,
      • Zhao S.
      • Borden M.
      • Bloch S.H.
      • Kruse D.
      • Ferrara K.W.
      • Dayton P.A.
      Radiation-force assisted targeting facilitates ultrasonic molecular imaging.
      ,
      • Rychak J.J.
      • Klibanov A.L.
      • Hossack J.A.
      Acoustic radiation force enhances targeted delivery of ultrasound contrast microbubbles: In vitro verification.
      ,
      • Lum A.F.H.
      • Borden M.A.
      • Dayton P.A.
      • Kruse D.E.
      • Simon S.I.
      • Ferrara K.W.
      Ultrasound radiation force enables targeted deposition of model drug carriers loaded on microbubbles.
      and
      • Frinking P.J.A.
      • Tardy I.
      • Théraulaz M.
      • Arditi M.
      • Powers J.
      • Pochon S.
      • Tranquart F.
      Effects of acoustic radiation force on the binding efficiency of BR55, a VEGFR2-specific ultrasound contrast agent.